Jack and Bill are playing a computer game. Jack has 90 points and loses 6 points each round. Bill has 50 points and earns 2 points each round.

Question

The relationships between rounds played, x, and points, y, for Jack and Bill are graphed on the coordinate plane below. What is the point of intersection?
x
y
Jack
Bill
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
Rounds
Points
The point of intersection is (
,
).
Excellent!
Now that you have found the point of intersection, (5, 60), let’s check that work algebraically. Since Jack and Bill 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, 90 − 6x and 50 + 2x, equal to one another. Do that now.
x
y
Jack
Bill
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
Rounds
Points

90 − 6x
=
50 + 2x
Good work!
You just found the point of intersection on the graph. Now, let's check that work algebraically. The expression for Jack's points has been set equal to the expression for Bill's points. Solve the equation to find the value of x.
Jack's points = Bill's points
90−6x
=
50+2x

x
=

Answer 1

To find the value of \(x\) where Jack's points equal Bill's points, we start with the equation you provided:

\[ 90 - 6x = 50 + 2x \]

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

1. First, we add \(6x\) to both sides to move the \(x\) terms to one side:

\[ 90 = 50 + 2x + 6x \]

This simplifies to:

\[ 90 = 50 + 8x \]

2. Next, we subtract \(50\) from both sides:

\[ 90 - 50 = 8x \]

This gives us:

\[ 40 = 8x \]

3. Now, we divide both sides by \(8\):

\[ x = \frac{40}{8} \]

Thus,

\[ x = 5 \]

Now that we have \(x\), we can substitute \(x = 5\) back into either Jack or Bill's point equations to find the number of points at that number of rounds.

For Jack:

\[ y = 90 - 6(5) = 90 - 30 = 60 \]

For Bill:

\[ y = 50 + 2(5) = 50 + 10 = 60 \]

Both calculations confirm that at \(x = 5\), \(y = 60\).

So, the solution to your question is:

The value of \(x\) is \(5\).