## 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:

1. Elimination
2. Matrices
3. Substitution
4. 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 $$3$$.

### 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 \$$

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}} $$\ \Rightarrow \ \color{red}{ y = \frac{-126 }{ -21} = 6}$$$$\ \Rightarrow \ \color{red}{ 7x \ + \ 5(6) = 58 }$$ $$\ \Rightarrow \ \color{red}{ x = \frac{58 \ - \ (30)}{7} = 4}$$

## Systems of Equations Word Problems Quiz

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