Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Mrs. Chen is shopping for school supplies with her children. Anita selected $2$ one-inch binders and $5$ two-inch binders, which cost a total of $\$29$ . Kevin selected $2$ one-inch binders and $2$ two-inch binders, which cost a total of $\$14$ . How much does each size of binder cost? $\newline$ A one-inch binder costs $\$$ _, and a two-inch binder costs $\$$ _.

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Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Mrs. Chen is shopping for school supplies with her children. Anita selected

2

one-inch binders and

5

two-inch binders, which cost a total of

\$29

. Kevin selected

2

one-inch binders and

2

two-inch binders, which cost a total of

\$14

. How much does each size of binder cost?

\newline

A one-inch binder costs

\$

_____, and a two-inch binder costs

\$

_____.

Form Equations: Let $x$ represent the cost of a one-inch binder and $y$ represent the cost of a two-inch binder. We can write two equations based on the information given: $\newline$ For Anita's purchase: $\newline$ $2x + 5y = 29$ $\newline$ For Kevin's purchase: $\newline$ $2x + 2y = 14$ $\newline$ We will use these two equations to form a system of equations that we can solve using the elimination method.
Elimination Method: To use the elimination method, we want to eliminate one of the variables by making the coefficients of that variable the same in both equations. We can do this by multiplying the second equation by $-1$ so that when we add the two equations, the $x$ terms will cancel out. $\newline$ Multiplying the second equation by $-1$ gives us: $\newline$ $-2x - 2y = -14$ $\newline$ Now we add this equation to the first equation: $\newline$ $(2x + 5y) + (-2x - 2y) = 29 + (-14)$
Combine Equations: After adding the equations, the $x$ terms cancel out and we are left with: $5y - 2y = 29 - 14$ Simplifying the left side gives us $3y$ , and simplifying the right side gives us $15$ : $3y = 15$ Now we divide both sides by $3$ to solve for $y$ : $y = \frac{15}{3}$ $y = 5$ So, the cost of a two-inch binder is $\$5$ .
Solve for y: Now that we know the cost of a two-inch binder, we can substitute $y = 5$ into one of the original equations to find the cost of a one-inch binder. We'll use the second equation for this: $\newline$ $2x + 2y = 14$ $\newline$ Substitute $y = 5$ into the equation: $\newline$ $2x + 2(5) = 14$ $\newline$ $2x + 10 = 14$ $\newline$ Now we subtract $10$ from both sides to solve for $x$ : $\newline$ $2x = 14 - 10$ $\newline$ $2x = 4$ $\newline$ Divide both sides by $2$ to find the value of $x$ : $\newline$ $2x + 2y = 14$ $1$ $\newline$ $2x + 2y = 14$ $2$ $\newline$ So, the cost of a one-inch binder is $2x + 2y = 14$ $3$ .