Tyler and Emma both track the number of miles they run each week. Tyler was already running 10 miles per week, and started running 2 more miles each week. Emma was already running 30 miles per week, and started running 3 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 Tyler and Emma run each week, are graphed on the coordinate plane below. What is the point of intersection?
Tyler
Emma
0
1
2
3
4
5
6
7
8
9
10
0
3
6
9
12
15
18
21
24
27
30
Time (weeks)
Number of miles
The point of intersection is (
,
).
Good work!
Now that you have found the point of intersection, (4, 18), let’s check that work algebraically. The equations for the relationship between weeks and miles for Tyler and Emma are shown below. Since the number of miles Tyler runs will be equal to the number of miles Emma runs at the solution to this system, you can set the expression for the number of miles they each run, 10+2x and 30−3x, equal to one another. Do that now.
Tyler
Emma
0
1
2
3
4
5
6
7
8
9
10
0
3
6
9
12
15
18
21
24
27
30
Time (weeks)
Number of miles

10+2x
=
30−3x
Good work!
Let’s solve the equation for x, the number of weeks when Tyler and Emma 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, 3x, while keeping the equation balanced?
Tyler’s miles = Emma's miles
10+2x = 30−3x

Add 3x to each side

Multiply each side by 3x

Subtract 3x from each side
You got it!
We can add 3x on both sides to get rid of the 3x on the right while keeping the equation balanced. Record adding 3x on both sides of the equation.
Tyler's miles
=
Emma's miles

10+2x
=
30−3x
+

3x
+

3x
Good work!
Add 3x on both sides and rewrite the equation to represent the remaining values.
Tyler's miles
=
Emma's miles

10+2x
=
30−3x
+
3x
+
3x

=
Great job!
We have all of the variables on the left. Now, keep going. What move can you make next to get rid of 10 on the left so that the x term, 5x, is alone on the left?
Tyler's miles
=
Emma's miles

10+2x
=
30−3x
+
3x
+
3x

10+5x
=
30

Subtract 10 from both sides

Add 5x on both sides.

Add 10 to both sides

Answer 1

Great job! The next step is indeed to isolate the \( x \) term on the left side of the equation by removing the constant. You can do that by subtracting 10 from both sides of the equation. Here's how it looks:

\[ 10 + 5x = 30 \]

Now, let's subtract 10 from both sides:

\[ 10 + 5x - 10 = 30 - 10 \]

This simplifies to:

\[ 5x = 20 \]

Now you can solve for \( x \) by dividing both sides by 5:

\[ x = \frac{20}{5} \]

Thus:

\[ x = 4 \]

This confirms that Tyler and Emma run the same number of miles after 4 weeks.

Now we can substitute \( x = 4 \) back into the original equations to find the number of miles they run at that point.

For Tyler:

\[ 10 + 2(4) = 10 + 8 = 18 \text{ miles} \]

For Emma:

\[ 30 - 3(4) = 30 - 12 = 18 \text{ miles} \]

So the point of intersection, where both run the same number of miles, is \( (4, 18) \).

To summarize, the point of intersection is:

\[ (4, 18) \]

Well done!