Jiro and Lane are playing a computer game. Jiro has 50 points and loses 2 points each round. Lane has 30 points and earns 2 points each round.

Question

The relationships between rounds played, x, and points, y, for Jiro and Lane are graphed on the coordinate plane below. What is the point of intersection?
x
y
Jiro
Lane
0
1
2
3
4
5
6
7
8
9
10
0
5
10
15
20
25
30
35
40
45
50
Rounds
Points
The point of intersection is (
,
).
You got it!
Now that you have found the point of intersection, (5, 40), let’s check that work algebraically. Since Jiro and Lane 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, 50 − 2x and 30 + 2x, equal to one another. Do that now.
x
y
Jiro
Lane
0
1
2
3
4
5
6
7
8
9
10
0
5
10
15
20
25
30
35
40
45
50
Rounds
Points

50 − 2x
=
30 + 2x
Great job!
You just found the point of intersection on the graph. Now, let's check that work algebraically. The expression for Jiro's points has been set equal to the expression for Lane's points. Solve the equation to find the value of x.
Jiro's points = Lane's points
50−2x
=
30+2x

x
=

Answer 1

To solve the equation \( 50 - 2x = 30 + 2x \) for \( x \), follow these steps:

1. First, add \( 2x \) to both sides:

\[ 50 - 2x + 2x = 30 + 2x + 2x \]

This simplifies to:

\[ 50 = 30 + 4x \]

2. Next, subtract 30 from both sides:

\[ 50 - 30 = 4x \]

This simplifies to:

\[ 20 = 4x \]

3. Now, divide both sides by 4:

\[ x = \frac{20}{4} \]

This gives:

\[ x = 5 \]

So the value of \( x \) is \( 5 \).

Now, let's find the corresponding \( y \) value for either Jiro or Lane at \( x = 5 \):

For Jiro's points:

\[ y = 50 - 2(5) = 50 - 10 = 40 \]

For Lane's points:

\[ y = 30 + 2(5) = 30 + 10 = 40 \]

Both give \( y = 40 \).

Thus, the point of intersection is \( (5, 40) \).

To summarize:

\( x = 5 \) and the point of intersection is \( (5, 40) \).