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

Question

Sidney and Emilio, the boy she was babysitting, were playing basketball together. Her score was 18 points, and his score was 19 points. Sidney wanted to make the game more fair, so she called a time-out and modified the rules a bit. Sidney explained that, for the rest of the game, she would get 3 points per basket, and Emilio would get 2 points per basket. Then they played a bit longer. After the time-out, they both made the same number of baskets and ended up with a tied score. How many baskets did each person make after the time out? How many points did each person have at the end?

Sidney and Emilio each made
baskets after the time-out, for a score of
.

Answer 1

Let's denote the number of baskets Sidney made after the time-out as \( x \) and the number of baskets Emilio made after the time-out as \( x \) as well since both made the same number of baskets.

Initially, Sidney had 18 points and Emilio had 19 points. After the time-out, Sidney scores 3 points for each basket, and Emilio scores 2 points for each basket. The system of equations can be set up as follows:

1. Sidney's final score: \( 18 + 3x \)
2. Emilio's final score: \( 19 + 2x \)

Since they end up with tied scores, we can equate their final scores:

\[ 18 + 3x = 19 + 2x \]

Now, let's solve for \( x \):

1. Subtract \( 2x \) from both sides:

\[ 18 + 3x - 2x = 19 \] \[ 18 + x = 19 \]

2. Subtract 18 from both sides:

\[ x = 1 \]

This means both Sidney and Emilio made \( 1 \) basket after the time-out.

Now, we can calculate their final scores:

• Sidney's final score:

\[ 18 + 3 \cdot 1 = 18 + 3 = 21 \]

• Emilio's final score:

\[ 19 + 2 \cdot 1 = 19 + 2 = 21 \]

Thus, both ended up with a score of 21 points.

To summarize, Sidney and Emilio each made 1 basket after the time-out, for a score of 21.