Marni writes this system to determine how much of each ingredient Alyssa plans to buy. She uses x to represent cups of frozen fruit puree and y to represent cups of yogurt.

Question

x + y = 6,
2x + 3y = 14

To solve this system using the linear combination method, Marni multiplies the first equation by −2 to eliminate the x variable.

−2x − 2y = −12
2x + 3y = 14

Finish solving the system to determine how many cups of fruit puree and yogurt Alyssa will buy.

Marni determined that Alyssa plans to purchase
cups of fruit puree and
cups of yogurt.

Answer 1

To continue solving the system after Marni multiplied the first equation by -2, we have the following two equations:

1. \(-2x - 2y = -12\)
2. \(2x + 3y = 14\)

Next, we will add these two equations together to eliminate the \(x\) variable:

\[ (-2x - 2y) + (2x + 3y) = -12 + 14 \]

This simplifies to:

\[ (-2y + 3y) = 2 \]

So, we have:

\[ y = 2 \]

Now that we have the value of \(y\), we can substitute \(y = 2\) back into the first equation to find \(x\):

\[ x + y = 6 \]

Substituting \(y\):

\[ x + 2 = 6 \]

Subtracting 2 from both sides:

\[ x = 4 \]

Thus, Marni determined that Alyssa plans to purchase:

• 4 cups of frozen fruit puree
• 2 cups of yogurt

So the final answer is: Alyssa plans to purchase 4 cups of fruit puree and 2 cups of yogurt.