Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Carmen likes to make desserts for bake sales. Last month, she made $3$ batches of brownies and $3$ batches of cookies, which called for $18$ eggs total. The month before, she baked $3$ batches of brownies and $1$ batch of cookies, which required a total of $10$ eggs. How many eggs did Carmen use for a batch of each dessert? $\newline$ Carmen uses $\_$ eggs to make a batch of brownies and $\_$ eggs to make a batch of cookies.

View step-by-step help

Home

Math Problems

Algebra 2

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

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

3

batches of brownies and

3

batches of cookies, which called for

18

eggs total. The month before, she baked

3

batches of brownies and

1

batch of cookies, which required a total of

10

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

\newline

Carmen uses

\_

eggs to make a batch of brownies and

\_

eggs to make a batch of cookies.

Define Variables: Let's denote the number of eggs used for a batch of brownies as $b$ and the number of eggs used for a batch of cookies as $c$ . From the first situation, we have the equation $3b + 3c = 18$ .
Form Equations: From the second situation, we have the equation $3b + 1c = 10$ .
Elimination Method: We now have a system of two equations: $\newline$ $1$ . $3b + 3c = 18$ $\newline$ $2$ . $3b + 1c = 10$ $\newline$ We will use elimination to solve this system. To eliminate $b$ , we can subtract the second equation from the first.
Solve for c: Subtracting the second equation from the first gives us $3b + 3c - (3b + 1c) = 18 - 10$ , which simplifies to $2c = 8$ .
Substitute $c$ into Equation: Dividing both sides of $2c = 8$ by $2$ gives us $c = 4$ . This means Carmen uses $4$ eggs to make a batch of cookies.
Solve for $b$ : Now we substitute $c = 4$ into the second equation $3b + 1c = 10$ to find $b$ . This gives us $3b + 1(4) = 10$ , which simplifies to $3b + 4 = 10$ .
Solve for b: Now we substitute $c = 4$ into the second equation $3b + 1c = 10$ to find $b$ . This gives us $3b + 1(4) = 10$ , which simplifies to $3b + 4 = 10$ . Subtracting $4$ from both sides of $3b + 4 = 10$ gives us $3b = 6$ .
Solve for b: Now we substitute $c = 4$ into the second equation $3b + 1c = 10$ to find b. This gives us $3b + 1(4) = 10$ , which simplifies to $3b + 4 = 10$ . Subtracting $4$ from both sides of $3b + 4 = 10$ gives us $3b = 6$ . Dividing both sides of $3b = 6$ by $3$ gives us $b = 2$ . This means Carmen uses $3b + 1c = 10$ $0$ eggs to make a batch of brownies.