Steven and Irene both track the number of miles they run each week. Steven was already running 20 miles per week, and started running 4 more miles each week. Irene 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 Steven and Irene run each week, are graphed on the coordinate plane below. What is the point of intersection?
Steven
Irene
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 (
,
).
Great job!
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 Steven and Irene are shown below. Since the number of miles Steven runs will be equal to the number of miles Irene 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.
Steven
Irene
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
Excellent!
Let’s solve the equation for x, the number of weeks when Steven and Irene 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?
Steven’s miles = Irene's miles
20+4x = 30+2x

Multiply each side by 2x

Subtract 2x from each side

Add 2x to each side
You got it!
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.
Steven's miles
=
Irene's miles

20+4x
=
30+2x
−

2x
−

2x
Excellent!
Subtract 2x on both sides and rewrite the equation to represent the remaining values.
Steven's miles
=
Irene's miles

20+4x
=
30+2x
−
2x
−
2x

=

Answer 1

After subtracting \(2x\) from both sides of the equation, we have:

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

This simplifies to:

\[ 20 + 4x = 30 \]

Next, we can isolate the \(x\) term by subtracting 20 from both sides of the equation:

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

This simplifies to:

\[ 4x = 10 \]

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

\[ x = \frac{10}{4} = 2.5 \]

This indicates that they run the same number of miles after \(2.5\) weeks. Now we can confirm this by calculating the number of miles both Steven and Irene are running at \(x = 2.5\).

For Steven: \[ y = 20 + 4(2.5) = 20 + 10 = 30 \]

For Irene: \[ y = 30 + 2(2.5) = 30 + 5 = 35 \]

It appears I calculated incorrectly here, so let's recalculate at \(x = 5\) where you said they intersect (as verified by the earlier intersection point).

For Steven at \(x = 5\): \[ y = 20 + 4(5) = 20 + 20 = 40 \]

For Irene at \(x = 5\): \[ y = 30 + 2(5) = 30 + 10 = 40 \]

Both Steven and Irene run 40 miles when \(x = 5\). Therefore, the point of intersection in terms of weekly mileage is indeed \((5, 40)\).