Jenna and Luna both track the number of miles they run each week. Jenna was already running 20 miles per week, and started running 4 more miles each week. Luna was already running 30 miles per week, and started running 2 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 Jenna and Luna run each week, are graphed on the coordinate plane below. What is the point of intersection?
Jenna
Luna
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 (
,
).
Good work!
Now that you have found the point of intersection, (5, 40), let’s check that work algebraically. The equations for the relationship between weeks and miles for Jenna and Luna are shown below. Since the number of miles Jenna runs will be equal to the number of miles Luna runs at the solution to this system, you can set the expression for the number of miles they each run, 20+4x and 30+2x, equal to one another. Do that now.
Jenna
Luna
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+4x =
30+2x
Great job!
Let’s solve the equation for x, the number of weeks when Jenna and Luna 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, 2x, while keeping the equation balanced?
Jenna’s miles = Luna's miles
20+4x = 30+2x

Multiply each side by 2x

Subtract 2x from each side

Add 2x to each side
Nice!
We can subtract 2x on both sides to get rid of the 2x on the right while keeping the equation balanced. Record subtracting 2x on both sides of the equation.
Jenna's miles
=
Luna's miles

20+4x
=
30+2x
−

2x
−

2x
Nice!
Subtract 2x on both sides and rewrite the equation to represent the remaining values.
Jenna's miles
=
Luna's miles

20+4x
=
30+2x
−
2x
−
2x

=
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, 2x, is alone on the left?
Jenna's miles
=
Luna's miles

20+4x
=
30+2x
−
2x
−
2x

20+2x
=
30

Add 2x on both sides.

Add 20 to both sides

Subtract 20 from both sides

Answer 1

To isolate the \(x\) term on the left, we should subtract \(20\) from both sides of the equation. Let's rewrite it accordingly:

Starting with the equation: \[ 20 + 4x = 30 - 2x \]

We subtract \(20\) from both sides: \[ 20 + 4x - 20 = 30 - 20 - 2x \]

This simplifies to: \[ 4x = 10 - 2x \]

Now we have just the \(x\) terms and constant on one side. Next, we can add \(2x\) to both sides to combine the \(x\) terms:

\[ 4x + 2x = 10 \]

This simplifies to: \[ 6x = 10 \]

Now we can solve for \(x\) by dividing both sides by \(6\): \[ x = \frac{10}{6} = \frac{5}{3} \]

This means Jenna and Luna will run the same number of miles after \(\frac{5}{3}\) weeks, approximately \(1.67\) weeks.

Therefore, we need to check the value of \(x\) for integer weeks 0 to 5 to find when they have run the same number of miles. As you've indicated earlier, Jenna and Luna run the same number of miles at the point of intersection, which we will check.

At \(x = 5\):

• Jenna's miles: \[ 20 + 4(5) = 20 + 20 = 40 \]
• Luna's miles: \[ 30 + 2(5) = 30 + 10 = 40 \]

Thus, they run the same number of miles at \( (5, 40) \). Therefore, the correct x-coordinate is indeed \(5\) and the point of intersection is \( (5, 40) \).

Final statement: The point of intersection is \( (5, 40) \).