Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Evelyn likes to make desserts for bake sales. Last month, she made $2$ batches of brownies and $1$ batch of cookies, which called for $9$ eggs total. The month before, she baked $2$ batches of brownies and $2$ batches of cookies, which required a total of $12$ eggs. How many eggs did Evelyn use for a batch of each dessert? $\newline$ Evelyn uses $\_$ eggs to make a batch of brownies and $\_$ eggs to make a batch of cookies.

Full solution

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

\newline

Evelyn likes to make desserts for bake sales. Last month, she made

2

batches of brownies and

1

batch of cookies, which called for

9

eggs total. The month before, she baked

2

batches of brownies and

2

batches of cookies, which required a total of

12

eggs. How many eggs did Evelyn use for a batch of each dessert?

\newline

Evelyn uses

\_

eggs to make a batch of brownies and

\_

eggs to make a batch of cookies.

Define variables: Let's define two variables: let $b$ be the number of eggs used for a batch of brownies, and let $c$ be the number of eggs used for a batch of cookies. We can write two equations based on the information given: $\newline$ $1$ . For the first month: $2b + 1c = 9$ (two batches of brownies and one batch of cookies used $9$ eggs) $\newline$ $2$ . For the month before: $2b + 2c = 12$ (two batches of brownies and two batches of cookies used $12$ eggs)
Write equations: Now, we will use the elimination method to solve the system of equations. We can do this by subtracting the first equation from the second equation to eliminate the variable $b$ . $\newline$ $(2b + 2c) - (2b + 1c) = 12 - 9$ $\newline$ This simplifies to: $\newline$ $2c - 1c = 3$
Use elimination method: Solving the simplified equation gives us the number of eggs used for a batch of cookies: $c = 3$
Solve for cookies: Now that we know the value of $c$ , we can substitute it back into one of the original equations to find the value of $b$ . Let's use the first equation: $\newline$ $2b + 1c = 9$ $\newline$ Substituting $c = 3$ , we get: $\newline$ $2b + 1(3) = 9$
Substitute back: Solving for $b$ , we subtract $3$ from both sides of the equation: $\newline$ $2b + 3 = 9$ $\newline$ $2b = 9 - 3$ $\newline$ $2b = 6$
Solve for brownies: Dividing both sides by $2$ gives us the number of eggs used for a batch of brownies: $\newline$ $b = \frac{6}{2}$ $\newline$ $b = 3$