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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The Maynard family and the Boyer family are seeing a movie together in the theater. At the concession stand, Mr. Maynard paid $\$18$ for $1$ large popcorn and $3$ large drinks that his family will share. Mrs. Boyer bought $3$ large popcorns and $2$ large drinks and paid $\$26$ . How much does each item cost? $\newline$ A large popcorn costs $\$\_\_\_\_$ and a large drink costs $\$\_\_\_\_$ .

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Solve a system of equations using elimination: word problems

Full solution

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

\newline

The Maynard family and the Boyer family are seeing a movie together in the theater. At the concession stand, Mr. Maynard paid

\$18

for

1

large popcorn and

3

large drinks that his family will share. Mrs. Boyer bought

3

large popcorns and

2

large drinks and paid

\$26

. How much does each item cost?

\newline

A large popcorn costs

\$\_\_\_\_

and a large drink costs

\$\_\_\_\_

.

Define variables: Let's define the variables for the cost of the items. $\newline$ Let $x$ be the cost of a large popcorn. $\newline$ Let $y$ be the cost of a large drink.
Write Maynard's equation: Write the equation for the Maynard family's purchase. $\newline$ $1$ large popcorn ( $x$ ) + $3$ large drinks ( $y$ ) = $\$18$ $\newline$ So, the equation is: $x + 3y = 18$
Write Boyer's equation: Write the equation for the Boyer family's purchase. $\newline$ $3$ large popcorns ( $x$ ) + $2$ large drinks ( $y$ ) = $\$26$ $\newline$ So, the equation is: $3x + 2y = 26$
Eliminate variable $x$ : We have the system of equations: $\newline$ $x + 3y = 18$ $\newline$ $3x + 2y = 26$ $\newline$ We need to eliminate one of the variables. Let's eliminate $x$ .
Multiply first equation: Multiply the first equation by $-3$ to eliminate $x$ . $\newline$ $-3(x + 3y) = -3(18)$ $\newline$ $-3x - 9y = -54$
Add equations: Add the new equation to the second equation to eliminate $x$ . $(-3x - 9y) + (3x + 2y) = -54 + 26$ $-3$ x + $3$ x - $9$ y + $2$ y = $-54$ + $26$ \) $0x - 7y = -28$
Solve for y: Solve for y. $\newline$ $-7y = -28$ $\newline$ $y = \frac{-28}{-7}$ $\newline$ $y = 4$
Substitute $y$ into first equation: Substitute $y = 4$ into the first equation to solve for $x$ .
$x + 3(4) = 18$
$x + 12 = 18$
$x = 18 - 12$
$x = 6$
Final solution: We found: $\newline$ $x = 6$ $\newline$ $y = 4$ $\newline$ A large popcorn costs $\$6$ and a large drink costs $\$4$ .

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