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?

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
You got it!
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.
Jenna's miles
=
Luna's miles

20+4x
=
30+2x

2x

2x

20 + 2x
=
30


20


20
Excellent!
Subtract 20 from both sides and rewrite the equation to represent the remaining values.
Jenna's miles
=
Luna's miles

20+4x
=
30+2x

2x

2x

20 + 2x
=
30

20

20

=
Nice!
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.
Jenna's miles
=
Luna's miles

20+4x
=
30+2x

2x

2x

20 + 2x
=
30

20

20

2x
=
10

x
=

1 answer

Great job working through the algebra! Let's summarize and carry on to find the value of \( x \).

Starting from the equation after your last operation:

\[ 20 + 2x = 10 \]

You need to isolate \( x \). You can do this by first subtracting 20 from both sides of the equation:

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

This simplifies to:

\[ 2x = -10 \]

Now, to solve for \( x \), divide both sides by 2:

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

Thus:

\[ x = -5 \]

However, this result doesn’t match with the previous findings. It looks like we may have made a mistake or misrepresented our equation earlier. Let's double-check:

Starting from:

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

We subtract \( 2x \) from both sides:

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

That gives:

\[ 20 + 2x = 30 \]

Now we subtract 20 from both sides correctly:

\[ 2x = 30 - 20 \]

Simplifying that gives:

\[ 2x = 10 \]

Finally, dividing by 2 yields:

\[ x = 5 \]

So the correct value for \( x \) is:

\[ x = 5 \]

At \( x = 5 \) weeks, Jenna and Luna both run the same number of miles, which is confirmed to be 40 miles (as noted in the point of intersection being (5, 40)).

Excellent work through the problem! The final answer is:

\[ x = 5 \]