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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Samantha and her friend Mabel are each baking apple pies and tarts for a bake sale, using the same recipes. Samantha baked $7$ apple pies and $8$ apple tarts, using a total of $88$ apples. Mabel made $1$ apple pie and $8$ apple tarts, which used $40$ apples. How many apples does each dessert require? $\newline$ An apple pie uses $\_$ apples and an apple tart requires $\_$ apples.

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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

Samantha and her friend Mabel are each baking apple pies and tarts for a bake sale, using the same recipes. Samantha baked

7

apple pies and

8

apple tarts, using a total of

88

apples. Mabel made

1

apple pie and

8

apple tarts, which used

40

apples. How many apples does each dessert require?

\newline

An apple pie uses

\_

apples and an apple tart requires

\_

apples.

Define Variables: Define the variables for the number of apples used in each dessert. $\newline$ Let $x$ be the number of apples used in one apple pie. $\newline$ Let $y$ be the number of apples used in one apple tart.
Write Equations: Write the system of equations based on the given information. $\newline$ Samantha's baking: $7$ pies and $8$ tarts used $88$ apples. $\newline$ Mabel's baking: $1$ pie and $8$ tarts used $40$ apples. $\newline$ This gives us the two equations: $\newline$ $7x + 8y = 88$ (Equation $1$ ) $\newline$ $1x + 8y = 40$ (Equation $2$ )
Use Elimination: Use elimination to solve the system of equations. $\newline$ We will eliminate $x$ by subtracting Equation $2$ from Equation $1$ . $\newline$ $(7x + 8y) - (1x + 8y) = 88 - 40$ $\newline$ $7x - 1x + 8y - 8y = 88 - 40$ $\newline$ $6x = 48$
Solve for x: Solve for x. $\newline$ Divide both sides of the equation by $6$ to find the value of $x$ . $\newline$ $\frac{6x}{6} = \frac{48}{6}$ $\newline$ $x = 8$
Substitute and Solve: Substitute the value of $x$ into one of the original equations to solve for $y$ . Using Equation $2$ : $1x + 8y = 40$ $1(8) + 8y = 40$ $8 + 8y = 40$ $8y = 40 - 8$ $8y = 32$
Solve for y: Solve for y. $\newline$ Divide both sides of the equation by $8$ to find the value of $y$ . $\newline$ $\frac{8y}{8} = \frac{32}{8}$ $\newline$ $y = 4$
Final Answer: State the final answer. $\newline$ An apple pie uses $8$ apples and an apple tart requires $4$ apples.

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