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
You got it!
We can subtract 6x on both sides to get rid of the 6x on the right while keeping the equation balanced. Record subtracting 6x on both sides of the equation.
Irene's miles
=
Charlie's miles

30+4x
=
20+6x
−

6x
−

6x
Great job!
Subtract 6x on both sides and rewrite the equation to represent the remaining values.
Irene's miles
=
Charlie's miles

30+4x
=
20+6x
−
6x
−
6x

=

Answer 1

After subtracting \(6x\) from both sides of the equation, we can rewrite it as follows:

\[ 30 + 4x = 20 \]

Now, we are left with the equation:

\[ 30 + 4x = 20 \]

Next, we can isolate the term \(4x\) by subtracting 30 from both sides:

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

This simplifies to:

\[ 4x = -10 \]

Now we can solve for \(x\) by dividing both sides by 4:

\[ x = \frac{-10}{4} \]

Which simplifies to:

\[ x = -2.5 \]

Since this is not a valid answer in the context of running miles over weeks, it seems I miscalculated or made an incorrect substitution somewhere.

Let me go back to the step right after subtracting 6x from both sides to clarify. Please check:

After the initial subtraction:

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

Subtract \(6x\) from both sides:

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

So this becomes:

\[ 30 - 2x = 20 \]

Now, to isolate \(-2x\), we can subtract 30 from both sides:

\[ -2x = 20 - 30 \]

Thus we have:

\[ -2x = -10 \]

Now, divide both sides by -2:

\[ x = 5 \]

So we find that \(x = 5\), which confirms the x-coordinate of the point of intersection.

The point of intersection is indeed \( (5, 50) \).

To summarize:

The equations we started with were

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

After solving we confirmed:

\[ x = 5 \]

Hence, the point of intersection is:

\[ (5, 50) \]

Irene and Charlie both run 50 miles at week 5.