Irene and Charlie both track the number of miles they run each week. Irene was already running 30 miles per week, and started running 4 more miles each week. Charlie was already running 20 miles per week, and started running 6 more miles each week. Is there a time when they run the same number of miles each week?

Question

The relationships between x, the number of weeks, and y, the number of miles Irene and Charlie run each week, are graphed on the coordinate plane below. What is the point of intersection?
Irene
Charlie
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
Time (weeks)
Number of miles
The point of intersection is (
,
).
Nice!
Now that you have found the point of intersection, (5, 50), let’s check that work algebraically. The equations for the relationship between weeks and miles for Irene and Charlie are shown below. Since the number of miles Irene runs will be equal to the number of miles Charlie runs at the solution to this system, you can set the expression for the number of miles they each run, 30+4x and 20+6x, equal to one another. Do that now.
Irene
Charlie
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
Time (weeks)
Number of miles

30+4x
=
20+6x
Excellent!
Let’s solve the equation for x, the number of weeks when Irene and Charlie run the same number of miles, to confirm that it’s the same as the x-coordinate of the point of intersection. Since there are x terms on both sides of the equation, let’s get rid of the x term from one of the sides. What move can you make to get rid of the x term on the right side of the equation, 6x, while keeping the equation balanced?
Irene’s miles = Charlie's miles
30+4x = 20+6x

Subtract 6x from each side

Add 6x to each side

Multiply each side by 6x

Answer 1

To solve the equation \( 30 + 4x = 20 + 6x \), we want to get rid of the \( 6x \) term on the right side of the equation. The best move to keep the equation balanced is to subtract \( 6x \) from each side of the equation.

So, the equation becomes:

\[ 30 + 4x - 6x = 20 \]

Now, simplifying the left side gives us:

\[ 30 - 2x = 20 \]

Next, we can solve for \( x \) by isolating it. If we subtract 30 from both sides, we get:

\[ -2x = 20 - 30 \] \[ -2x = -10 \]

Now, divide both sides by -2:

\[ x = \frac{-10}{-2} = 5 \]

Thus, \( x = 5 \) weeks, which matches the \( x \)-coordinate of the point of intersection.

So our final confirmation is \( x = 5 \).

The point of intersection is \( (5, 50) \), confirming that after 5 weeks, both Irene and Charlie will be running 50 miles each week.