Question

Jackie goes to the mall every weekend with her friends. Last week, the group spent $28.00 on 2 salads and 4 fruit smoothies. This week, the group spent $38.00 on 3 salads and 5 fruit smoothies. Let x represent the cost of a salad and let y represent the cost of a smoothie.

2x + 4y = 28

3x + 5y = 38

If you multiply the first equation by –3, what could you multiply the second equation by to eliminate the x-variable when adding the equations?
What is the cost of a salad?
What is the cost of a smoothie?

Answer 1

To eliminate the \( x \)-variable when adding the equations, we first multiply the first equation by \(-3\):

\[
-3(2x + 4y) = -3(28)
\]
This results in:
\[
-6x - 12y = -84
\]

Now, the second equation is:
\[
3x + 5y = 38
\]

We want to eliminate \( x \) when we add the two equations together. To do this, we need to multiply the second equation by 2. This will make the coefficient of \( x \) in the second equation match the coefficient in the modified first equation:

\[
2(3x + 5y) = 2(38)
\]
This results in:
\[
6x + 10y = 76
\]

Now we can add the two equations together:

\[
(-6x - 12y) + (6x + 10y) = -84 + 76
\]
The \( x \) terms will cancel:

\[
(-12y + 10y) = -84 + 76
\]
This simplifies to:
\[
-2y = -8
\]
Now, solve for \( y \):

\[
y = \frac{-8}{-2} = 4
\]

Now that we have \( y \), we can substitute \( y = 4 \) back into one of the original equations to find \( x \). We can use the first equation:

\[
2x + 4y = 28
\]
Substituting for \( y \):
\[
2x + 4(4) = 28
\]
This simplifies to:
\[
2x + 16 = 28
\]
Now, solve for \( x \):

\[
2x = 28 - 16
\]
\[
2x = 12
\]
\[
x = \frac{12}{2} = 6
\]

Therefore, the costs are:
- The cost of a salad (\( x \)) is $6.00.
- The cost of a smoothie (\( y \)) is $4.00.