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

=

Answer 1

Let's start with the equation:

\[ 20 + 4x = 30 + 2x \]

Now, we'll subtract \( 2x \) from both sides to get:

\[ 20 + 4x - 2x = 30 + 2x - 2x \]

This simplifies to:

\[ 20 + 2x = 30 \]

Now, you can subtract 20 from both sides:

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

This simplifies to:

\[ 2x = 10 \]

Next, divide both sides by 2:

\[ x = 5 \]

So, the number of weeks when Jenna and Luna run the same number of miles is \( x = 5 \).

Now, to find the number of miles they are both running at this point, substitute \( x = 5 \) back into either person's running equation. We'll use Jenna's equation:

\[ y = 20 + 4(5) \] \[ y = 20 + 20 \] \[ y = 40 \]

Now substitute \( x = 5 \) into Luna's equation to confirm:

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

Both calculations confirm that they run the same number of miles after 5 weeks, which is 40 miles.

The point of intersection is therefore \( (5, 40) \).

So, the final representation of the equation after subtracting \( 2x \) is:

\[ 20 + 2x = 30 \]