Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Yesterday a chef used $40$ eggs to make $10$ chocolate souffles and $10$ lemon meringue pies. The day before, he made $2$ chocolate souffles and $6$ lemon meringue pies, which used $16$ eggs. How many eggs does each dessert require? $\newline$ A chocolate souffle requires $\_$ eggs and a lemon meringue pie requires $\_$ eggs.

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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

Yesterday a chef used

40

eggs to make

10

chocolate souffles and

10

lemon meringue pies. The day before, he made

2

chocolate souffles and

6

lemon meringue pies, which used

16

eggs. How many eggs does each dessert require?

\newline

A chocolate souffle requires

\_

eggs and a lemon meringue pie requires

\_

eggs.

Define Variables: Let's define two variables: let $x$ be the number of eggs required for one chocolate souffle, and $y$ be the number of eggs required for one lemon meringue pie. We can then write two equations based on the information given: $\newline$ Equation $1$ (from yesterday's production): $10x + 10y = 40$ $\newline$ Equation $2$ (from the day before yesterday's production): $2x + 6y = 16$ $\newline$ We will use these equations to form a system of equations that we can solve using the elimination method.
Form Equations: To use the elimination method, we want to eliminate one of the variables by making the coefficients of either $x$ or $y$ the same in both equations. We can multiply the entire second equation by $5$ to match the coefficients of $x$ in the first equation: $\newline$ $5(2x + 6y) = 5(16)$ $\newline$ $10x + 30y = 80$ $\newline$ Now we have a new system of equations: $\newline$ $10x + 10y = 40$ (Equation $1$ ) $\newline$ $10x + 30y = 80$ (New Equation $2$ )
Elimination Method: Next, we subtract Equation $1$ from the new Equation $2$ to eliminate the $x$ variable: $\newline$ $(10x + 30y) - (10x + 10y) = 80 - 40$ $\newline$ $10x + 30y - 10x - 10y = 80 - 40$ $\newline$ $20y = 40$ $\newline$ Now we can solve for $y$ by dividing both sides by $20$ : $\newline$ $\frac{20y}{20} = \frac{40}{20}$ $\newline$ $y = 2$ $\newline$ So, each lemon meringue pie requires $2$ eggs.
Substitute and Solve: Now that we have the value for $y$ , we can substitute it back into one of the original equations to solve for $x$ . We'll use Equation $1$ : $\newline$ $10x + 10y = 40$ $\newline$ $10x + 10(2) = 40$ $\newline$ $10x + 20 = 40$ $\newline$ Subtract $20$ from both sides to solve for $x$ : $\newline$ $10x = 40 - 20$ $\newline$ $10x = 20$ $\newline$ Divide both sides by $10$ : $\newline$ $x$ $0$ $\newline$ $x$ $1$ $\newline$ So, each chocolate souffle also requires $x$ $2$ eggs.