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Mathematics LibreTexts

1.20: Word Problems for Linear Equations

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Word problems are important applications of linear equations. We start with examples of translating an English sentence or phrase into an algebraic expression.

Example 18.1

Translate the phrase into an algebraic expression:

a) Twice a variable is added to 4

Solution: We call the variable \(x .\) Twice the variable is \(2 x .\) Adding \(2 x\) to 4 gives:

\[4 + 2x\nonumber\]

b) Three times a number is subtracted from 7.

Solution: Three times a number is \(3 x .\) We need to subtract \(3 x\) from 7. This means:\

\[7-3 x\nonumber\]

c) 8 less than a number.

Solution: The number is denoted by \(x .8\) less than \(x\) mean, that we need to subtract 8 from it. We get:

\[x-8\nonumber\]

For example, 8 less than 10 is \(10-8=2\).

d) Subtract \(5 p^{2}-7 p+2\) from \(3 p^{2}+4 p\) and simplify.

Solution: We need to calculate \(3 p^{2}+4 p\) minus \(5 p^{2}-7 p+2:\)

\[\left(3 p^{2}+4 p\right)-\left(5 p^{2}-7 p+2\right)\nonumber\]

Simplifying this expression gives:

\[\left(3 p^{2}+4 p\right)-\left(5 p^{2}-7 p+2\right)=3 p^{2}+4 p-5 p^{2}+7 p-2 =-2 p^{2}+11 p-2\nonumber\]

e) The amount of money given by \(x\) dimes and \(y\) quarters.

Solution: Each dime is worth 10 cents, so that this gives a total of \(10 x\) cents. Each quarter is worth 25 cents, so that this gives a total of \(25 y\) cents. Adding the two amounts gives a total of

\[10 x+25 y \text{ cents or } .10x + .25y \text{ dollars}\nonumber\]

Now we deal with word problems that directly describe an equation involving one variable, which we can then solve.

Example 18.2

Solve the following word problems:

a) Five times an unknown number is equal to 60. Find the number.

Solution: We translate the problem to algebra:

\[5x = 60\nonumber\]

We solve this for \(x\) :

\[x=\frac{60}{5}=12\nonumber\]

b) If 5 is subtracted from twice an unknown number, the difference is \(13 .\) Find the number.

Solution: Translating the problem into an algebraic equation gives:

\[2x − 5 = 13\nonumber\]

We solve this for \(x\). First, add 5 to both sides.

\[2x = 13 + 5, \text{ so that } 2x = 18\nonumber\]

Dividing by 2 gives \(x=\frac{18}{2}=9\).

c) A number subtracted from 9 is equal to 2 times the number. Find the number.

Solution: We translate the problem to algebra.

\[9 − x = 2x\nonumber\]

We solve this as follows. First, add \(x\) :

\[9 = 2x + x \text{ so that } 9 = 3x\nonumber\]

Then the answer is \(x=\frac{9}{3}=3\)

d) Multiply an unknown number by five is equal to adding twelve to the unknown number. Find the number.

Solution: We have the equation:

\[5x = x + 12.\nonumber\]

Subtracting \(x\) gives

\[4x = 12.\nonumber\]

Dividing both sides by 4 gives the answer: \(x=3\).

e) Adding nine to a number gives the same result as subtracting seven from three times the number. Find the number.

Solution: Adding 9 to a number is written as \(x+9,\) while subtracting 7 from three times the number is written as \(3 x-7\). We therefore get the equation:

\[x + 9 = 3x − 7.\nonumber\]

We solve for \(x\) by adding 7 on both sides of the equation:

\[x + 16 = 3x.\nonumber\]

Then we subtract \(x:\)

\[16 = 2x.\nonumber\]

After dividing by \(2,\) we obtain the answer \(x=8\)

The following word problems consider real world applications. They require to model a given situation in the form of an equation.

Example 18.3

a) Due to inflation, the price of a loaf of bread has increased by \(5 \%\). How much does the loaf of bread cost now, when its price was \(\$ 2.40\) last year?

Solution: We calculate the price increase as \(5 \% \cdot \$ 2.40 .\) We have

\[5 \% \cdot 2.40=0.05 \cdot 2.40=0.1200=0.12\nonumber\]

We must add the price increase to the old price.

\[2.40+0.12=2.52\nonumber\]

The new price is therefore \(\$ 2.52\).

b) To complete a job, three workers get paid at a rate of \(\$ 12\) per hour. If the total pay for the job was \(\$ 180,\) then how many hours did the three workers spend on the job?

Solution: We denote the number of hours by \(x\). Then the total price is calculated as the price per hour \((\$ 12)\) times the number of workers times the number of hours \((3) .\) We obtain the equation

\[12 \cdot 3 \cdot x=180\nonumber\]

Simplifying this yields

\[36 x=180\nonumber\]

Dividing by 36 gives

\[x=\frac{180}{36}=5\nonumber\]

Therefore, the three workers needed 5 hours for the job.

c) A farmer cuts a 300 foot fence into two pieces of different sizes. The longer piece should be four times as long as the shorter piece. How long are the two pieces?

\[x+4 x=300\nonumber\]

Combining the like terms on the left, we get

\[5 x=300\nonumber\]

Dividing by 5, we obtain that

\[x=\frac{300}{5}=60\nonumber\]

Therefore, the shorter piece has a length of 60 feet, while the longer piece has four times this length, that is \(4 \times 60\) feet \(=240\) feet.

d) If 4 blocks weigh 28 ounces, how many blocks weigh 70 ounces?

Solution: We denote the weight of a block by \(x .\) If 4 blocks weigh \(28,\) then a block weighs \(x=\frac{28}{4}=7\)

How many blocks weigh \(70 ?\) Well, we only need to find \(\frac{70}{7}=10 .\) So, the answer is \(10 .\)

Note You can solve this problem by setting up and solving the fractional equation \(\frac{28}{4}=\frac{70}{x}\). Solving such equations is addressed in chapter 24.

e) If a rectangle has a length that is three more than twice the width and the perimeter is 20 in, what are the dimensions of the rectangle?

Solution: We denote the width by \(x\). Then the length is \(2 x+3\). The perimeter is 20 in on one hand and \(2(\)length\()+2(\)width\()\) on the other. So we have

\[20=2 x+2(2 x+3)\nonumber\]

Distributing and collecting like terms give

\[20=6 x+6\nonumber\]

Subtracting 6 from both sides of the equation and then dividing both sides of the resulting equation by 6 gives:

\[20-6=6 x \Longrightarrow 14=6 x \Longrightarrow x=\frac{14}{6} \text { in }=\frac{7}{3} \text { in }=2 \frac{1}{3} \text { in. }\nonumber\]

f) If a circle has circumference 4in, what is its radius?

Solution: We know that \(C=2 \pi r\) where \(C\) is the circumference and \(r\) is the radius. So in this case

\[4=2 \pi r\nonumber\]

Dividing both sides by \(2 \pi\) gives

\[r=\frac{4}{2 \pi}=\frac{2}{\pi} \text { in } \approx 0.63 \mathrm{in}\nonumber\]

g) The perimeter of an equilateral triangle is 60 meters. How long is each side?

Solution: Let \(x\) equal the side of the triangle. Then the perimeter is, on the one hand, \(60,\) and on other hand \(3 x .\) So \(3 x=60\) and dividing both sides of the equation by 3 gives \(x=20\) meters.

h) If a gardener has \(\$ 600\) to spend on a fence which costs \(\$ 10\) per linear foot and the area to be fenced in is rectangular and should be twice as long as it is wide, what are the dimensions of the largest fenced in area?

Solution: The perimeter of a rectangle is \(P=2 L+2 W\). Let \(x\) be the width of the rectangle. Then the length is \(2 x .\) The perimeter is \(P=2(2 x)+2 x=6 x\). The largest perimeter is \(\$ 600 /(\$ 10 / f t)=60\) ft. So \(60=6 x\) and dividing both sides by 6 gives \(x=60 / 6=10\). So the dimensions are 10 feet by 20 feet.

i) A trapezoid has an area of 20.2 square inches with one base measuring 3.2 in and the height of 4 in. Find the length of the other base.

Solution: Let \(b\) be the length of the unknown base. The area of the trapezoid is on the one hand 20.2 square inches. On the other hand it is \(\frac{1}{2}(3.2+b) \cdot 4=\) \(6.4+2 b .\) So

\[20.2=6.4+2 b\nonumber\]

Multiplying both sides by 10 gives

\[202=64+20 b\nonumber\]

Subtracting 64 from both sides gives

\[b=\frac{138}{20}=\frac{69}{10}=6.9 \text { in }\nonumber\]

and dividing by 20 gives

Exit Problem

Write an equation and solve: A car uses 12 gallons of gas to travel 100 miles. How many gallons would be needed to travel 450 miles?

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\(\textbf{1)}\) Joe and Steve are saving money. Joe starts with $105 and saves $5 per week. Steve starts with $5 and saves $15 per week. After how many weeks do they have the same amount of money? Show Equations \(y= 5x+105,\,\,\,y=15x+5\) Show Answer 10 weeks ($155)

\(\textbf{2)}\) mike and sarah collect rocks. together they collected 50 rocks. mike collected 10 more rocks than sarah. how many rocks did each of them collect show equations \(m+s=50,\,\,\,m=s+10\) show answer mike collected 30 rocks, sarah collected 20 rocks., \(\textbf{3)}\) in a classroom the ratio of boys to girls is 2:3. there are 25 students in the class. how many are girls show equations \(b+g=50,\,\,\,3b=2g\) show answer 15 girls (10 boys), \(\textbf{4)}\) kyle makes sandals at home. the sandal making tools cost $100 and he spends $10 on materials for each sandal. he sells each sandal for $30. how many sandals does he have to sell to break even show equations \(c=10x+100,\,\,\,r=30x\) show answer 5 sandals ($150), \(\textbf{5)}\) molly is throwing a beach party. she still needs to buy beach towels and beach balls. towels are $3 each and beachballs are $4 each. she bought 10 items in total and it cost $34. how many beach balls did she get show equations show answer 4 beachballs (6 towels), \(\textbf{6)}\) anna volunteers at a pet shelter. they have cats and dogs. there are 36 pets in total at the shelter, and the ratio of dogs to cats is 4:5. how many cats are at the shelter show equations \(c+d=40,\,\,\,5d=4c\) show answer 20 cats (16 dogs), \(\textbf{7)}\) a store sells oranges and apples. oranges cost $1.00 each and apples cost $2.00 each. in the first sale of the day, 15 fruits were sold in total, and the price was $25. how many of each type of frust was sold show equations \(o+a=15,\,\,\,1o+2a=25\) show answer 10 apples and 5 oranges, \(\textbf{8)}\) the ratio of red marbles to green marbles is 2:7. there are 36 marbles in total. how many are red show equations \(r+g=36,\,\,\,7r=2g\) show answer 8 red marbles (28 green marbles), \(\textbf{9)}\) a tennis club charges $100 to join the club and $10 for every hour using the courts. write an equation to express the cost \(c\) in terms of \(h\) hours playing tennis. show equation the equation is \(c=10h+100\), \(\textbf{10)}\) emma and liam are saving money. emma starts with $80 and saves $10 per week. liam starts with $120 and saves $6 per week. after how many weeks will they have the same amount of money show equations \(e = 10x + 80,\,\,\,l = 6x + 120\) show answer 10 weeks ($180 each), \(\textbf{11)}\) mark and lisa collect stamps. together they collected 200 stamps. mark collected 40 more stamps than lisa. how many stamps did each of them collect show equations \(m + l = 200,\,\,\,m = l + 40\) show answer mark collected 120 stamps, lisa collected 80 stamps., \(\textbf{12)}\) in a classroom, the ratio of boys to girls is 3:5. there are 40 students in the class. how many are boys show equations \(b + g = 40,\,\,\,5b = 3g\) show answer 15 boys (25 girls), \(\textbf{13)}\) lisa is selling handmade jewelry. the materials cost $60, and she sells each piece for $20. how many pieces does she have to sell to break even show equations \(c=60,\,\,\,r=20x\) show answer 3 pieces, \(\textbf{14)}\) tom is buying books and notebooks for school. books cost $15 each, and notebooks cost $3 each. he bought 12 items in total, and it cost $120. how many notebooks did he buy show equations \(b + n = 12,\,\,\,15b+3n=120\) show answer 5 notebooks (7 books), \(\textbf{15)}\) emily volunteers at an animal shelter. they have rabbits and guinea pigs. there are 36 animals in total at the shelter, and the ratio of guinea pigs to rabbits is 4:5. how many guinea pigs are at the shelter show equations \(r + g = 36,\,\,\,5g=4r\) show answer 16 guinea pigs (20 rabbits), \(\textbf{16)}\) mike and sarah are going to a theme park. mike’s ticket costs $40, and sarah’s ticket costs $30. they also bought $20 worth of food. how much did they spend in total show equations \(m + s + f = t,\,\,\,m=40,\,\,\,s=30,\,\,\,f=20\) show answer they spent $90 in total., \(\textbf{17)}\) the ratio of red marbles to blue marbles is 2:3. there are 50 marbles in total. how many are blue show equations \(r + b = 50,\,\,\,3r=2b\) show answer 30 blue marbles (20 red marbles), \(\textbf{18)}\) a pizza restaurant charges $12 for a large pizza and $8 for a small pizza. if a customer buys 5 pizzas in total, and it costs $52, how many large pizzas did they buy show equations \(l + s = 5,\,\,\,12l+8s=52\) show answer they bought 3 large pizzas (2 small pizzas)., \(\textbf{19)}\) the area of a rectangle is 48 square meters. if the length is 8 meters, what is the width of the rectangle show equations \(a=l\times w,\,\,\,l=8,\,\,\,a=48\) show answer the width is 6 meters., \(\textbf{20)}\) two numbers have a sum of 50. one number is 10 more than the other. what are the two numbers show equations \(x+y=50,\,\,\,x=y+10\) show answer the numbers are 30 and 20., \(\textbf{21)}\) a store sells jeans for $40 each and t-shirts for $20 each. in the first sale of the day, they sold 8 items in total, and the price was $260. how many of each type of item was sold show equations \(j+t=8,\,\,\,40j+20t=260\) show answer 5 jeans and 3 t-shirts were sold., \(\textbf{22)}\) the ratio of apples to carrots is 3:4. there are 28 fruits in total. how many are apples show equations \(\)a+c=28,\,\,\,4a=3c show answer there are 12 apples and 16 carrots., \(\textbf{23)}\) a phone plan costs $30 per month, and there is an additional charge of $0.10 per minute for calls. write an equation to express the cost \(c\) in terms of \(m\) minutes. show equation the equation is \(\)c=30+0.10m, \(\textbf{24)}\) a triangle has a base of 8 inches and a height of 6 inches. calculate its area. show equations \(a=0.5\times b\times h,\,\,\,b=8,\,\,\,h=6\) show answer the area is 24 square inches., \(\textbf{25)}\) a store sells shirts for $25 each and pants for $45 each. in the first sale of the day, 4 items were sold, and the price was $180. how many of each type of item was sold show equations \(t+p=4,\,\,\,25t+45p=180\) show answer 0 shirts and 4 pants were sold., \(\textbf{26)}\) a garden has a length of 12 feet and a width of 10 feet. calculate its area. show equations \(a=l\times w,\,\,\,l=12,\,\,\,w=10\) show answer the area is 120 square feet., \(\textbf{27)}\) the sum of two consecutive odd numbers is 56. what are the two numbers show equations \(x+y=56,\,\,\,x=y+2\) show answer the numbers are 27 and 29., \(\textbf{28)}\) a toy store sells action figures for $15 each and toy cars for $5 each. in the first sale of the day, 10 items were sold, and the price was $110. how many of each type of item was sold show equations \(a+c=10,\,\,\,15a+5c=110\) show answer 6 action figures and 4 toy cars were sold., \(\textbf{29)}\) a bakery sells pie for $2 each and cookies for $1 each. in the first sale of the day, 14 items were sold, and the price was $25. how many of each type of item was sold show equations \(p+c=14,\,\,\,2p+c=25\) show answer 11 pies and 3 cookies were sold., \(\textbf{for 30-33}\) two car rental companies charge the following values for x miles. car rental a: \(y=3x+150 \,\,\) car rental b: \(y=4x+100\), \(\textbf{30)}\) which rental company has a higher initial fee show answer company a has a higher initial fee, \(\textbf{31)}\) which rental company has a higher mileage fee show answer company b has a higher mileage fee, \(\textbf{32)}\) for how many driven miles is the cost of the two companies the same show answer the companies cost the same if you drive 50 miles., \(\textbf{33)}\) what does the \(3\) mean in the equation for company a show answer for company a, the cost increases by $3 per mile driven., \(\textbf{34)}\) what does the \(100\) mean in the equation for company b show answer for company b, the initial cost (0 miles driven) is $100., \(\textbf{for 35-39}\) andy is going to go for a drive. the formula below tells how many gallons of gas he has in his car after m miles. \(g=12-\frac{m}{18}\), \(\textbf{35)}\) what does the \(12\) in the equation represent show answer andy has \(12\) gallons in his car when he starts his drive., \(\textbf{36)}\) what does the \(18\) in the equation represent show answer it takes \(18\) miles to use up \(1\) gallon of gas., \(\textbf{37)}\) how many miles until he runs out of gas show answer the answer is \(216\) miles, \(\textbf{38)}\) how many gallons of gas does he have after 90 miles show answer the answer is \(7\) gallons, \(\textbf{39)}\) when he has \(3\) gallons remaining, how far has he driven show answer the answer is \(162\) miles, \(\textbf{for 40-42}\) joe sells paintings. each month he makes no commission on the first $5,000 he sells but then makes a 10% commission on the rest., \(\textbf{40)}\) find the equation of how much money x joe needs to sell to earn y dollars per month. show answer the answer is \(y=.1(x-5,000)\), \(\textbf{41)}\) how much does joe need to sell to earn $10,000 in a month. show answer the answer is \($105,000\), \(\textbf{42)}\) how much does joe earn if he sells $45,000 in a month show answer the answer is \($4,000\), see related pages\(\), \(\bullet\text{ word problems- linear equations}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- averages}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- consecutive integers}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- distance, rate and time}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- break even}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- ratios}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- age}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ word problems- mixtures and concentration}\) \(\,\,\,\,\,\,\,\,\), linear equations are a type of equation that has a linear relationship between two variables, and they can often be used to solve word problems. in order to solve a word problem involving a linear equation, you will need to identify the variables in the problem and determine the relationship between them. this usually involves setting up an equation (or equations) using the given information and then solving for the unknown variables . linear equations are commonly used in real-life situations to model and analyze relationships between different quantities. for example, you might use a linear equation to model the relationship between the cost of a product and the number of units sold, or the relationship between the distance traveled and the time it takes to travel that distance. linear equations are typically covered in a high school algebra class. these types of problems can be challenging for students who are new to algebra, but they are an important foundation for more advanced math concepts. one common mistake that students make when solving word problems involving linear equations is failing to set up the problem correctly. it’s important to carefully read the problem and identify all of the relevant information, as well as any given equations or formulas that you might need to use. other related topics involving linear equations include graphing and solving systems. understanding linear equations is also useful for applications in fields such as economics, engineering, and physics., about andymath.com, andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. if you have any requests for additional content, please contact andy at [email protected] . he will promptly add the content. topics cover elementary math , middle school , algebra , geometry , algebra 2/pre-calculus/trig , calculus and probability/statistics . in the future, i hope to add physics and linear algebra content. visit me on youtube , tiktok , instagram and facebook . andymath content has a unique approach to presenting mathematics. the clear explanations, strong visuals mixed with dry humor regularly get millions of views. we are open to collaborations of all types, please contact andy at [email protected] for all enquiries. to offer financial support, visit my patreon page. let’s help students understand the math way of thinking thank you for visiting. how exciting.

linear equations practice 100 problems

Linear Equations in the real world

A page on how to find the equation and how to graph real world applications of linear equations.

  • Interpreting Graphs of Real World Linear Equations
  • Worksheet on real world linear equations (worksheet version of this web page)
  • slope of a line
  • y-intercept
  • interactive linear equation
  • equation given slope and a point
  • equation from graph of a line
  • real world applications

A cab company charges a $3 boarding rate in addition to its meter which is $2 for every mile. What is the equation of the line that represents this cab company's rate?

linear equations practice 100 problems

A cab company charges a $5 boarding rate in addition to its meter which is $3 for every mile. What is the equation of the line that represents this cab company's rate?

linear equations practice 100 problems

A cab company charges a $3 boarding rate in addition to its meter which is $½ for every mile. What is the equation of the line that represents this cab company's rate?

linear equations practice 100 problems

A cab company charges a $4 boarding rate in addition to its meter which is $ ¾ for every mile. What is the equation of the line that represents this cab company's rate?

linear equations practice 100 problems

A cab company does not charge a boarding fee but then has a meter of $4 an hour. What equation represents this cab company's rate?

linear equations practice 100 problems

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linear equations practice 100 problems

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Unit 4: Systems of linear equations

Writing & graphing systems of linear equations.

  • Systems of equations: trolls, tolls (1 of 2) (Opens a modal)
  • Systems of equations: trolls, tolls (2 of 2) (Opens a modal)
  • Testing a solution to a system of equations (Opens a modal)
  • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1 (Opens a modal)
  • Systems of equations with graphing: exact & approximate solutions (Opens a modal)
  • Setting up a system of equations from context example (pet weights) (Opens a modal)
  • Setting up a system of linear equations example (weight and price) (Opens a modal)
  • Interpreting points in context of graphs of systems (Opens a modal)
  • Solutions of systems of equations Get 3 of 4 questions to level up!
  • Systems of equations with graphing Get 3 of 4 questions to level up!
  • Creating systems in context Get 3 of 4 questions to level up!
  • Interpret points relative to a system Get 3 of 4 questions to level up!

Solving systems by substitution

  • Systems of equations with substitution: potato chips (Opens a modal)
  • Systems of equations with substitution: -3x-4y=-2 & y=2x-5 (Opens a modal)
  • Substitution method review (systems of equations) (Opens a modal)
  • Systems of equations with substitution Get 3 of 4 questions to level up!

Solving systems by elimination (part 1)

  • Why can we subtract one equation from the other in a system of equations? (Opens a modal)
  • Worked example: equivalent systems of equations (Opens a modal)
  • Worked example: non-equivalent systems of equations (Opens a modal)
  • Reasoning with systems of equations (Opens a modal)
  • Equivalent systems of equations review (Opens a modal)
  • Reasoning with systems of equations Get 3 of 4 questions to level up!

Solving systems by elimination (part 2)

  • Systems of equations with elimination: King's cupcakes (Opens a modal)
  • Systems of equations with elimination: x-4y=-18 & -x+3y=11 (Opens a modal)
  • Systems of equations with elimination Get 3 of 4 questions to level up!

Solving systems by elimination (part 3)

  • Elimination strategies (Opens a modal)
  • Systems of equations with elimination: potato chips (Opens a modal)
  • Systems of equations with elimination (and manipulation) (Opens a modal)
  • Elimination method review (systems of linear equations) (Opens a modal)
  • Elimination strategies Get 3 of 4 questions to level up!
  • Systems of equations with elimination challenge Get 3 of 4 questions to level up!

Systems of linear equations and their solutions

  • Systems of equations number of solutions: fruit prices (1 of 2) (Opens a modal)
  • Systems of equations number of solutions: fruit prices (2 of 2) (Opens a modal)
  • Solutions to systems of equations: consistent vs. inconsistent (Opens a modal)
  • Solutions to systems of equations: dependent vs. independent (Opens a modal)
  • Number of solutions to a system of equations (Opens a modal)
  • Number of solutions to a system of equations graphically (Opens a modal)
  • Number of solutions to a system of equations algebraically (Opens a modal)
  • How many solutions does a system of linear equations have if there are at least two? (Opens a modal)
  • Number of solutions to system of equations review (Opens a modal)
  • Number of solutions to a system of equations graphically Get 3 of 4 questions to level up!
  • Number of solutions to a system of equations algebraically Get 3 of 4 questions to level up!

Linear Equations Questions

The linear equations questions and answers will assist students to understand the concepts better. Linear Equation is a topic that is covered in basically every class. The NCERT guidelines will be followed for preparing the questions. Linear Equations are used in mathematics as well as in everyday life. So, the basics of this concept must be grasped. For students of all levels, the problems here will include both the basics and more challenging problems. As a result, students will be able to use it to solve problems involving linear equations. Learn more about Linear Equations by clicking here .

Here, we’ll go through a variety of linear equations problems with solutions, based on various concepts.

Linear Equations Questions with Solutions

1. Write the statement as an equation: A number increased by 8 equals 15.

Given statement: A number increased by 8 equals 15.

Let the number be “x”.

So, x is increased by 8 means x + 8.

Hence, x increased by 8 equals 15 means x + 8 = 15, which is the equation for the given statement.

2. Write the statement for the given equation: 2x = 18.

Given equation: 2x = 18.

The statement for the given equation is “Twice the number x equals 18”.

3. Solve the equation: x + 3 = -2

Given equation: x + 3 = -2.

Now, keep the variables on one side and constants on the other side. Hence, the equation becomes,

Hence, the value of x is -5.

4. Verify that x = 4 is the root of the equation 3x/2 = 6.

To verify whether the given root is the solution of the given equation, substitute x = 4 in the equation 3x/2 = 6.

⇒ (3(4))/2 = 6

⇒ (12/2) = 6

Hence, x = 4 is the root of the equation 3x/2 = 6.

5. If 5 is added to twice a number, the result is 29. Determine the number.

The equation for the given statement is 5+2x = 29.

To find the number “x”, we have to solve the equation.

⇒ 2x = 29 – 5

Hence, the required number is 12.

6. If x = 2, then 2x – 5 = 7. Check whether the statement is true or false.

Given equation: 2x – 5 = 7

= 2(2) – 5

= 4 – 5 = -1

Hence, the given statement is false.

7. The sum of two consecutive numbers is 11. Find the numbers.

Let the number be x.

Hence, the two consecutive numbers are x and x+1.

According to the given statement, the equation becomes

⇒ x + x + 1 = 11

⇒ 2x + 1 = 11

⇒ x = 10/2 = 5

If x = 5, then x + 1 = 5 + 1 = 6

Hence, the two numbers are 5 and 6.

8. Express the equation x = 3y in the form of ax+by+c = 0 and find the values of a, b and c.

Given equation: x = 3y

We know that the standard form of linear equation in two variables is ax+by+c = 0 …(1)

Now, rearranging the given equation, we get

⇒ x – 3y = 0

This can be written as

⇒ 1(x) + (-3)y + (0)c = 0 …(2)

On comparing equation (1) and (2), we get

⇒ a = 1, b = -3 and c = 0.

9. Find three solutions for the equation 2x + y = 7.

To find the solutions for the equation 2x + y = 7, substitute different values for x.

When x = 0,

⇒ 2(0) + y = 7

Therefore, the solution is (0, 7).

When x = 1,

⇒ 2(1) + y = 7

⇒ y = 7 – 2

Hence, the solution is (1, 5).

When x = 2,

⇒ 2(2) + y = 7

⇒ 4 + y = 7

Hence, the solution is (2, 3).

Therefore, the three solutions are (0, 7), (1, 5) and (2, 3).

10. Solve the following equations using the substitution method:

3x + 4y = 10 and 2x – 2y = 2

3x + 4y = 10 …(1)

2x – 2y = 2 …(2)

Equation (2) can be written as:

2(x – y) = 2

x – y = 1

x = 1+y …(3)

Now, substitute (3) in (1), we get

3 (1+y) + 4y = 10

3 + 3y + 4y = 10

7y = 10 – 3

Hence, y = 1.

Now, substitute y = 1 in (3), we get

Hence, x = 2 and y = 1 are the solutions of the given equations.

Practice Questions

  • Write the statement as an equation: Twice a number subtracted from 19 is 11.
  • The sum of the two numbers is 30 and their ratio is 2: 3. Find the numbers.
  • If the point (3, 4) lies on the graph of equation 3y = ax + 7, determine the value of a.
  • Solve the equations using the elimination method: (x/2)+(2y/3) = -1 and x – (y/3) = 3.

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  4. Linear Equations Worksheets with Answer Key

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  6. Linear Equations Practice Activities by Math Giraffe

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VIDEO

  1. Math 8: Linear Tables to Linear Equations Practice!

  2. Math 8: Linear Tables to Linear Equations Practice!

  3. 1 4b 6 Solving Linear Equations

  4. 1 4b 1 Solving Linear Equations

  5. Practice set 1.3 Linear equation in two variables class 10

  6. Linear Equations Class 10

COMMENTS

  1. Algebra

    Here is a set of practice problems to accompany the Linear Equations section of the Solving Equations and Inequalities chapter of the notes for Paul Dawkins Algebra course at Lamar University.

  2. Linear equations, functions, & graphs

    Algebra (all content) 20 units · 412 skills. Unit 1 Introduction to algebra. Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs. Unit 4 Sequences. Unit 5 System of equations. Unit 6 Two-variable inequalities. Unit 7 Functions. Unit 8 Absolute value equations, functions, & inequalities.

  3. 8.E: Solving Linear Equations (Exercises)

    8.3 - Solve Equations with Variables and Constants on Both Sides. In the following exercises, solve the equations with constants on both sides. 8p + 7 = 47. 10w − 5 = 65. 3x + 19 = −47. 32 = −4 − 9n. In the following exercises, solve the equations with variables on both sides. 7y = 6y − 13. 5a + 21 = 2a.

  4. Linear equations in any form

    Algebra 1 > Forms of linear equations > Summary: Forms of two-variable linear equations Linear equations in any form Google Classroom You might need: Calculator 2 4 6 8 − 4 − 6 − 8 2 4 6 8 − 4 − 6 − 8 y x Write an equation that represents the line. Use exact numbers. Show Calculator Stuck? Do 4 problems

  5. Solve Linear Equations Practice

    Solve Linear Equations Practice - MathBitsNotebook (A1) Directions: Solve the following equations, for the indicated variable. Be careful! The students' choices may, or may not, be correct. 1. Solve for x: 3 x - 12 = 0. Choose: x = 4.

  6. Worksheets for linear equations

    Min: Max: Number of decimal digits used in the variable (s) and constant (s) Number of empty lines below the problem (workspace) Choose the types of equations generated for the worksheet. Choose AT LEAST one type. Type 1: one-step equations (the simplest possible, such as x + 6 = 19 or 6x = 17 or x/7 = 18) nonnegative solutions only

  7. Linear equations & graphs

    Quiz Unit test About this unit Let's explore different ways to find and visualize slopes and intercepts, and how these concepts can help us solve real-world problems. Two-variable linear equations intro Learn Two-variable linear equations intro Solutions to 2-variable equations Worked example: solutions to 2-variable equations

  8. Linear Equations Practice

    Practice Linear Equations, receive helpful hints, take a quiz, improve your math skills. ... Word Problems. ... Expand Factor Exponents Logarithms Radicals Complex Numbers Linear Equations Quadratic Equations Rational Equations Radical Equations Logarithmic Equations Exponential Equations Absolute Equations Polynomials Inequalities System of ...

  9. Linear Equations Practice Problems

    1. Equations & Inequalities Linear Equations Practice Problems 30 problems 1 PRACTICE PROBLEM For the statement given below, write if it is true or false. The equation 6x + 8 = 3x - 7 has a solution set as {-5}. 2 PRACTICE PROBLEM For the statement given below, write if it is true or false. 9x = 7x is a contradiction equation. 3 PRACTICE PROBLEM

  10. Solving Linear Equations: Practice Problems

    A linear equation is simply an algebraic expression that represents a line. These equations commonly contain one or two variables, usually x or y. These are called first-degree equations because ...

  11. 1.20: Word Problems for Linear Equations

    Example 18.2. Solve the following word problems: a) Five times an unknown number is equal to 60. Find the number. Solution: We translate the problem to algebra: 5x = 60 5 x = 60. We solve this for x x : x = 60 5 = 12 x = 60 5 = 12. b) If 5 is subtracted from twice an unknown number, the difference is 13.

  12. Algebra (Practice Problems)

    Algebra. Here are a set of practice problems for the Algebra notes. Click on the " Solution " link for each problem to go to the page containing the solution. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the ...

  13. Linear equations word problems

    Course: Algebra 1 > Unit 5 Math > Algebra 1 > Forms of linear equations > Intro to slope-intercept form Linear equations word problems Google Classroom Ever since Renata moved to her new home, she's been keeping track of the height of the tree outside her window. H represents the height of the tree (in centimeters), t years since Renata moved in.

  14. Problem Sets with Solutions

    MIT18_06SCF11_Ses3.5sol.pdf. pdf. MIT18_06SCF11_Ses3.6sol.pdf. pdf. MIT18_06SCF11_Ses3.7sol.pdf. MIT OpenCourseWare is a web based publication of virtually all MIT course content. OCW is open and available to the world and is a permanent MIT activity.

  15. Free Math Worksheets

    Khan Academy's 100,000+ free practice questions give instant feedback, don't need to be graded, and don't require a printer. Math Worksheets. Khan Academy. Math worksheets take forever to hunt down across the internet. Khan Academy is your one-stop-shop for practice from arithmetic to calculus. Math worksheets can vary in quality from ...

  16. PDF Precalculus: Linear Equations Practice Problems

    32. Find the equation of the line that passes through the points (2,0) and 3 2, 1 2 . 33. Find the equation of the line that passes through the point (4,3) and has slope m = −2. 34. Find the equation of the line that passes through the points (1,−8) and (2,−14). 35. Write the equation of the line given below. 36. Write the equation of the ...

  17. Word Problems Linear Equations

    Printable pages make math easy. Are you ready to be a mathmagician? \ (y= 5x+105,\,\,\,y=15x+5\) Show Answer 10 weeks ($155) \ (\textbf {2)}\) Mike and Sarah collect rocks. Together they collected 50 rocks. Mike collected 10 more rocks than Sarah. How many rocks did each of them collect? Show Equations \ (m+s=50,\,\,\,m=s+10\) Show Answer

  18. Linear Equations in the real world

    equation from graph of a line. real world applications. images. Problem 1. A cab company charges a $3 boarding rate in addition to its meter which is $2 for every mile. What is the equation of the line that represents this cab company's rate? Problem 2. A cab company charges a $5 boarding rate in addition to its meter which is $3 for every mile.

  19. Algebra

    Practice Problems; Assignment Problems; Show/Hide; Show all Solutions/Steps/etc. Hide all Solutions/Steps/etc. ... Linear Equations. Back to Problem List. 2. Solve the following equation and check your answer. \[2\left( {w + 3} \right) - 10 = 6\left( {32 - 3w} \right)\] ... It is important when doing this step to verify by plugging the solution ...

  20. Practice

    Practice - Graphing Linear Equations - MathBitsNotebook (A1) Directions: These questions pertain to slopes and graphs of lines. Choose the correct answer.

  21. Systems of linear equations

    Systems of equations: trolls, tolls (2 of 2) Testing a solution to a system of equations. Systems of equations with graphing: y=7/5x-5 & y=3/5x-1. Systems of equations with graphing: exact & approximate solutions. Setting up a system of equations from context example (pet weights) Setting up a system of linear equations example (weight and price)

  22. Linear(Simple) Equations: Very Difficult Problems with Solutions

    Problem 1 A number is equal to 7 times itself minus 18. Which is the number? Problem 2 A number is equal to 4 times this number less 75. What is the number? Problem 3 Two times a number, decreased by 12 equals three times the number, decreased by 15. Which is the number? Problem 4 Twice a number equals 5 times the same number plus 18.

  23. Linear Equations Questions with Solutions

    Solution: Given statement: A number increased by 8 equals 15. Let the number be "x". So, x is increased by 8 means x + 8. Hence, x increased by 8 equals 15 means x + 8 = 15, which is the equation for the given statement. 2. Write the statement for the given equation: 2x = 18. Solution: Given equation: 2x = 18.