How to Solve Systems of Equations Word Problems

System of Equations Word Problem?

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System of Equations

Generally, a system of linear equations is defined by two or more equations that contain the same variables. When we solve a system of two linear equations, we find the point of intersection of the two lines represented by the equations (if it exists). A system of linear equations can have a unique solution, infinitely many solutions, or no solutions. There are four primary methods to solve a system of linear equations:

Elimination
Matrices
Substitution
Graphing

In this article, we will learn about the elimination method. The elimination method involves adding, subtracting, or multiplying the equations in a linear system to eliminate one variable, making it easier to solve for the other variable(s).

Solving a system of equations word problem

Question: A class of $82$ students went on a field trip. A total of $7$ vehicles were taken, out of which some were cars and some buses. If each car holds $7$ students and each bus hold $18$ students, how many buses did they take?
Let’s consider the number of buses to be $x$ and the number of cars to be $y$ . So, it is clearly given that the total number of vehicles is $7$ , so, $x \ + \ y \ = \ 7$ .
Now, it is given that total number of students is $82$ and each car holds $7$ students and each bus holds $18$ . So, $18x \ + \ 7y \ = \ 82$ .
Now, let’s solve both these equations.

Firstly, we need to equate the coefficient of any term (be it $x$ or $y$ ) between the two equations. Here, we can see that if we multiply the first equation by $18$ on both sides, we get the same coefficient of $x$ as the second equation. So, let’s multiply the first equation by $18$ to get $18x \ + \ 18y \ = \ 126$ .
Now, our aim is to add or subtract both equations so that one variable gets eliminated. So, if we subtract both equations (equation $1 \ –$ equation $2$ ) we get $11y \ = \ 44$ , hence we get the value of $y \ = \ 4$ .
Now, if we put the value of $y$ in any equation (suppose equation $1$ ), we get $x \ + \ 4 \ = \ 7$ , hence we get the value of $x \ = \ 3$ .
So, we can say that this system of equations has the solution $x \ = \ 3$ and $y \ = \ 4$ . So, the number of buses is $3$ .

Free printable Worksheets

Exercises for Systems of Equations Word Problems

1) The equations of the two lines are $4x \ + \ 3y \ = \ 40$ and $7x \ + \ 6y \ = \ 73$ . Find the value of $x$ and $y$ in the solution for this system of equations.
$\begin{align} 4x \ + \ 3y &= 40 \\ 7x \ + \ 6y &= 73 \\ \hline \end{align}$ $\ \Rightarrow \$

2) A theater is selling tickets for a performance. Mr. Smith purchased $16$ senior tickets and $3$ child tickets for $$92$ for his friends and family. Mr. Jackson purchased $1$ senior tickets and $1$ child tickets for $$9$ . What is the price of a senior ticket?
$\begin{align} 16x \ + \ 3y &= 92 \\ x \ + \ y &= 9 \\ \hline \end{align}$ $\ \Rightarrow \$

3) At a store, Eva bought $1$ shirts and $2$ hats for $$11$ . Nicole bought $1$ same shirts and $1$ same hats for $$7$ . What is the price of each shirt?
$\begin{align} x \ + \ 2y &= 11 \\ x \ + \ y &= 7 \\ \hline \end{align}$ $\ \Rightarrow \$

4) The equations of the two lines are $4x \ + \ 3y \ = \ 40$ and $7x \ + \ 6y \ = \ 73$ . Find the value of $x$ and $y$ in the solution for this system of equations.
$\begin{align} 5x \ + \ 6y &= 50 \\ 3x \ + \ 5y &= 37 \\ \hline \end{align}$ $\ \Rightarrow \$

5) At a store, Eva bought $4$ shirts and $2$ hats for $$30$ . Nicole bought $1$ same shirts and $1$ same hats for $$10$ . What is the price of each shirt?
$\begin{align} 4x \ + \ 2y &= 30 \\ x \ + \ y &= 10 \\ \hline \end{align}$ $\ \Rightarrow \$

6) A class of $24$ students went on a field trip. They took $9$ vehicles, some cars, and some buses. If each car holds $2$ students and each bus hold $4$ students, how many buses did they take?
$\begin{align} 4x \ + \ 2y &= 24 \\ x \ + \ y &= 9 \\ \hline \end{align}$ $\ \Rightarrow \$

7) Emma and Sepehr are selling Chocolate Chip cookies and Oreo cookies Emma sold $3$ boxes of Chocolate Chip Cookies and $7$ boxes of Oreo cookies for a total of $$46$ . Sepehr sold $5$ boxes of Chocolate Chip Cookies and $6$ boxes of Oreo cookies for a total of $$54$ . Find the cost of one box of Chocolate Chip cookies.
$\begin{align} 3x \ + \ 7y &= 46 \\ 5x \ + \ 6y &= 54 \\ \hline \end{align}$ $\ \Rightarrow \$

8) A theater is selling tickets for a performance. Mr. Smith purchased $5$ senior tickets and $7$ child tickets for $$58$ for his friends and family. Mr. Jackson purchased $7$ senior tickets and $6$ child tickets for $$66$ . What is the price of a senior ticket?
$\begin{align} 5x \ + \ 7y &= 58 \\ 7x \ + \ 6y &= 66 \\ \hline \end{align}$ $\ \Rightarrow \$

9) The equations of the two lines are $x \ - \ y \ = \ 5$ and $x \ + \ y \ = \ 17$ . Find the value of $x$ and $y$ in the solution for this system of equations.
$\begin{align} x \ - \ y &= 5 \\ x \ + \ y &= 17 \\ \hline \end{align}$ $\ \Rightarrow \$

10) Emma and Sepehr are selling Chocolate Chip cookies and Oreo cookies Emma sold $7$ boxes of Chocolate Chip Cookies and $5$ boxes of Oreo cookies for a total of $$58$ . Sepehr sold $7$ boxes of Chocolate Chip Cookies and $2$ boxes of Oreo cookies for a total of $$40$ . Find the cost of one box of Chocolate Chip cookies.
$\begin{align} 7x \ + \ 5y &= 58 \\ 7x \ + \ 2y &= 40 \\ \hline \end{align}$ $\ \Rightarrow \$

Answers

1)The equations of the two lines are

$4x \ + \ 3y \ = \ 40$ and

$7x \ + \ 6y \ = \ 73$ . Find the value of

$x$ and

$y$ in the solution for this system of equations.

$\begin{align} 4x \ + \ 3y &= 40 \\ 7x \ + \ 6y &= 73 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -7(4x \ + \ 3y) &= -7(40) \\ 4(7x \ + \ 6y) &= 4(73) \\ \hline 3y &= 12 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{12 }{ 3} = 4}$

$\ \Rightarrow \ \color{red}{ 4x \ + \ 3(4) = 40 }$

$\ \Rightarrow \ \color{red}{ x = \frac{40 \ - \ (12)}{4} = 7}$

2)A theater is selling tickets for a performance. Mr. Smith purchased 16 senior tickets and 3 child tickets for $92 for his friends and family. Mr. Jackson purchased 1 senior tickets and 1 child tickets for $9. What is the price of a senior ticket?

$\begin{align} 16x \ + \ 3y &= 92 \\ x \ + \ y &= 9 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -1(16x \ + \ 3y) &= -1(92) \\ 16(x \ + \ y) &= 16(9) \\ \hline 13y &= 52 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{52 }{ 13} = 4}$

$\ \Rightarrow \ \color{red}{ 16x \ + \ 3(4) = 92 }$

$\ \Rightarrow \ \color{red}{ x = \frac{92 \ - \ (12)}{16} = 5}$

3)At a store, Eva bought

$1$ shirts and

$2$ hats for

$$11$ . Nicole bought

$1$ same shirts and

$1$ same hats for

$$7$ . What is the price of each shirt?

$\begin{align} x \ + \ 2y &= 11 \\ x \ + \ y &= 7 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -1(x \ + \ 2y) &= -1(11) \\ x \ + \ y &= 7 \\ \hline -y &= -4 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = 4}$

$\ \Rightarrow \ \color{red}{ x \ + \ 2(4) = 11 }$

$\ \Rightarrow \ \color{red}{ x = 11 \ - \ 8 = 3}$

4)The equations of the two lines are

$4x \ + \ 3y \ = \ 40$ and

$7x \ + \ 6y \ = \ 73$ . Find the value of

$x$ and

$y$ in the solution for this system of equations.

$\begin{align} 5x \ + \ 6y &= 50 \\ 3x \ + \ 5y &= 37 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -3(5x \ + \ 6y) &= -3(50) \\ 5(3x \ + \ 5y) &= 5(37) \\ \hline 7y &= 35 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{35 }{ 7} = 5}$

$\ \Rightarrow \ \color{red}{ 5x \ + \ 6(5) = 50 }$

$\ \Rightarrow \ \color{red}{ x = \frac{50 \ - \ (30)}{5} = 4}$

5)At a store, Eva bought

$4$ shirts and

$2$ hats for

$$30$ . Nicole bought

$1$ same shirts and

$1$ same hats for

$$10$ . What is the price of each shirt?

$\begin{align} 4x \ + \ 2y &= 30 \\ x \ + \ y &= 10 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -1(4x \ + \ 2y) &= -1(30) \\ 4(x \ + \ y) &= 4(10) \\ \hline 2y &= 10 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{10 }{ 2} = 5}$

$\ \Rightarrow \ \color{red}{ 4x \ + \ 2(5) = 30 }$

$\ \Rightarrow \ \color{red}{ x = \frac{30 \ - \ (10)}{4} = 5}$

6)A class of

$24$ students went on a field trip. They took

$9$ vehicles, some cars, and some buses. If each car holds

$2$ students and each bus hold

$4$ students, how many buses did they take?

$\begin{align} 4x \ + \ 2y &= 24 \\ x \ + \ y &= 9 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -1(4x \ + \ 2y) &= -1(24) \\ 4(x \ + \ y) &= 4(9) \\ \hline 2y &= 12 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{12 }{ 2} = 6}$

$\ \Rightarrow \ \color{red}{ 4x \ + \ 2(6) = 24 }$

$\ \Rightarrow \ \color{red}{ x = \frac{24 \ - \ (12)}{4} = 3}$

7)Emma and Sepehr are selling Chocolate Chip cookies and Oreo cookies Emma sold

$3$ boxes of Chocolate Chip Cookies and

$7$ boxes of Oreo cookies for a total of

$$46$ . Sepehr sold

$5$ boxes of Chocolate Chip Cookies and

$6$ boxes of Oreo cookies for a total of

$$54$ . Find the cost of one box of Chocolate Chip cookies.

$\begin{align} 3x \ + \ 7y &= 46 \\ 5x \ + \ 6y &= 54 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -5(3x \ + \ 7y) &= -5(46) \\ 3(5x \ + \ 6y) &= 3(54) \\ \hline -17y &= -68 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{-68 }{ -17} = 4}$

$\ \Rightarrow \ \color{red}{ 3x \ + \ 7(4) = 46 }$

$\ \Rightarrow \ \color{red}{ x = \frac{46 \ - \ (28)}{3} = 6}$

8)A theater is selling tickets for a performance. Mr. Smith purchased

$5$ senior tickets and

$7$ child tickets for

$$58$ for his friends and family. Mr. Jackson purchased

$7$ senior tickets and

$6$ child tickets for

$$66$ . What is the price of a senior ticket?

$\begin{align} 5x \ + \ 7y &= 58 \\ 7x \ + \ 6y &= 66 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -7(5x \ + \ 7y) &= -7(58) \\ 5(7x \ + \ 6y) &= 5(66) \\ \hline -19y &= -76 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{-76 }{ -19} = 4}$

$\ \Rightarrow \ \color{red}{ 5x \ + \ 7(4) = 58 }$

$\ \Rightarrow \ \color{red}{ x = \frac{58 \ - \ (28)}{5} = 6}$

9)The equations of the two lines are

$x \ - \ y \ = \ 5$ and

$x \ + \ y \ = \ 17$ . Find the value of

$x$ and

$y$ in the solution for this system of equations.

$\begin{align} x \ - \ y &= 5 \\ x \ + \ y &= 17 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -1(x \ - \ y) &= -1(5) \\ 1(x \ + \ y) &= 1(17) \\ \hline 2y &= 12 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{12 }{ 2} = 6}$

$\ \Rightarrow \ \color{red}{ 1x \ + \ (-1)(6) = 5 }$

$\ \Rightarrow \ \color{red}{ x = \frac{5 \ - \ (-6)}{1} = 11}$

10)Emma and Sepehr are selling Chocolate Chip cookies and Oreo cookies Emma sold

$7$ boxes of Chocolate Chip Cookies and

$5$ boxes of Oreo cookies for a total of

$$58$ . Sepehr sold

$7$ boxes of Chocolate Chip Cookies and

$2$ boxes of Oreo cookies for a total of

$$40$ . Find the cost of one box of Chocolate Chip cookies.

$\begin{align} 7x \ + \ 5y &= 58 \\ 7x \ + \ 2y &= 40 \\ \hline \end{align}$

$\ \Rightarrow \color{red}{ + \begin{align} -7(7x \ + \ 5y) &= -7(58) \\ 7(7x \ + \ 2y) &= 7(40) \\ \hline -21y &= -126 \end{align}}$

$\ \Rightarrow \ \color{red}{ y = \frac{-126 }{ -21} = 6}$

$\ \Rightarrow \ \color{red}{ 7x \ + \ 5(6) = 58 }$

$\ \Rightarrow \ \color{red}{ x = \frac{58 \ - \ (30)}{7} = 4}$

How to Solve Systems of Equations Word Problems

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Solving a system of equations word problem

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More Equations courses

New releases

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