Zoe and James both track the number of miles they run each week. Zoe was already running 20 miles per week, and started running 6 more miles each week. James was already running 40 miles per week, and started running 4 less 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 Zoe and James run each week, are graphed on the coordinate plane below. What is the point of intersection?
Zoe
James
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Time (weeks)
Number of miles
The point of intersection is (
,
).
You got it!
Now that you have found the point of intersection, (2, 32), let’s check that work algebraically. The equations for the relationship between weeks and miles for Zoe and James are shown below. Since the number of miles Zoe runs will be equal to the number of miles James runs at the solution to this system, you can set the expression for the number of miles they each run, 20+6x and 40−4x, equal to one another. Do that now.
Zoe
James
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Time (weeks)
Number of miles

20+6x =
40−4x
Excellent!
Let’s solve the equation for x, the number of weeks when Zoe and James 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, 4x, while keeping the equation balanced?
Zoe’s miles = James's miles
20+6x = 40−4x

Multiply each side by 4x

Subtract 4x from each side

Add 4x to each side
Nice!
We can add 4x on both sides to get rid of the 4x on the right while keeping the equation balanced. Record adding 4x on both sides of the equation.
Zoe's miles
=
James's miles

20+6x
=
40−4x
+

4x
+

4x
You got it!
Add 4x on both sides and rewrite the equation to represent the remaining values.
Zoe's miles
=
James's miles

20+6x
=
40−4x
+
4x
+
4x

=
Great job!
We have all of the variables on the left. Now, keep going. What move can you make next to get rid of 20 on the left so that the x term, 10x, is alone on the left?
Zoe's miles
=
James's miles

20+6x
=
40−4x
+
4x
+
4x

20+10x
=
40

Subtract 20 from both sides

Add 10x on both sides.

Add 20 to both sides
Nice!
You can subtract 20 on both sides so that the x term remains alone on the left, and the equation is balanced. Record that in the equation.
Zoe's miles
=
James's miles

20+6x
=
40−4x
+
4x
+
4x

20 + 10x
=
40
−

20
−

20
You got it!
Subtract 20 from both sides and rewrite the equation to represent the remaining values.
Zoe's miles
=
James's miles

20+6x
=
40−4x
+
4x
+
4x

20 + 10x
=
40
−
20
−
20

=
Excellent!
Keep going. Make a balanced move so that x is alone on the left, and then rewrite the equation to solve for the value of x.
Zoe's miles
=
James's miles

20+6x
=
40−4x
+
4x
+
4x

20 + 10x
=
40
−
20
−
20

10x
=
20

x
=

Answer 1

To solve for \( x \) in the equation \( 10x = 20 \), we can divide both sides by 10:

\[ x = \frac{20}{10} = 2 \]

So, after solving, we find that \( x = 2 \).

This means that Zoe and James both run the same number of miles after 2 weeks. To confirm, we can substitute \( x = 2 \) into the equations for the miles they run:

• For Zoe: \[ y = 20 + 6(2) = 20 + 12 = 32 \]

• For James: \[ y = 40 - 4(2) = 40 - 8 = 32 \]

Both values of \( y \) equal 32, confirming that the point of intersection is indeed \((2, 32)\).

So, the final answer to the question about the values of the point of intersection is:

The point of intersection is \( (2, 32) \).