Lee and Lucas are playing a computer game. Lee has 80 points and loses 12 points each round. Lucas has 20 points and earns 3 points each round.

Question

The relationships between rounds played, x, and points, y, for Lee and Lucas are graphed on the coordinate plane below. What is the point of intersection?
x
y
Lee
Lucas
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Rounds
Points
The point of intersection is (
,
).
Good work!
Now that you have found the point of intersection, (4, 32), let’s check that work algebraically. Since Lee and Lucas would have the same number of points at the solution to this system, you can set the expression for the number of points they each have, 80 − 12x and 20 + 3x, equal to one another. Do that now.
x
y
Lee
Lucas
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Rounds
Points

20 + 3x =
80 − 12x
Nice!
You just found the point of intersection on the graph. Now, let's check that work algebraically. The expression for Lee's points has been set equal to the expression for Lucas's points. Solve the equation to find the value of x.
Lee's points = Lucas's points
80−12x
=
20+3x

x
=

Answer 1

To find the value of \( x \), we will solve the equation:

\[ 80 - 12x = 20 + 3x \]

Now, let's isolate \( x \). First, we can combine like terms by moving \( 3x \) to the left side and \( 80 \) to the right side:

\[ 80 - 20 = 3x + 12x \]

This simplifies to:

\[ 60 = 15x \]

Now, to solve for \( x \), divide both sides by 15:

\[ x = \frac{60}{15} = 4 \]

Now that we have \( x = 4 \), we can substitute this back into either Lee's or Lucas's equation to find the corresponding number of points \( y \).

Let's use Lee's points equation:

\[ y = 80 - 12(4) \] \[ y = 80 - 48 = 32 \]

So, the point of intersection is \( (4, 32) \).

Thus, the value of \( x \) is:

\[ x = 4 \]