Dylan and Nora both track the number of miles they bike each week. Dylan was already biking 60 miles per week, and started biking 10 more miles each week. Nora was already biking 120 miles per week, and started biking 20 less miles each week. Is there a time when they bike the same number of miles each week?

Question

The relationships between x, the number of weeks, and y, the number of miles Dylan and Nora bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Dylan
Nora
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
200
Time (weeks)
Number of miles
The point of intersection is (
,
).
Good work!
Now that you have found the point of intersection, (2, 80), let’s check that work algebraically. The equations for the relationship between weeks and miles for Dylan and Nora are shown below. Since the number of miles Dylan bikes will be equal to the number of miles Nora bikes at the solution to this system, you can set the expression for the number of miles they each bike, 60+10x and 120−20x, equal to one another. Do that now.
Dylan
Nora
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
200
Time (weeks)
Number of miles

60+10x
=
120−20x
Excellent!
Let’s solve the equation for x, the number of weeks when Dylan and Nora bike 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, 20x, while keeping the equation balanced?
Dylan’s miles = Nora's miles
60+10x = 120−20x

Multiply each side by 20x

Add 20x to each side

Subtract 20x from each side

Answer 1

To solve the equation \(60 + 10x = 120 - 20x\) for \(x\) (the number of weeks when Dylan and Nora bike the same number of miles), we want to eliminate the \(20x\) term from the right side. To do this, we should add \(20x\) to each side of the equation so that we can combine like terms.

Here are the steps to solve the equation:

1. Start with the original equation: \[ 60 + 10x = 120 - 20x \]

2. Add \(20x\) to both sides: \[ 60 + 10x + 20x = 120 - 20x + 20x \]

3. Simplify both sides: \[ 60 + 30x = 120 \]

4. Next, subtract \(60\) from both sides: \[ 30x = 120 - 60 \]

5. Simplifying that gives: \[ 30x = 60 \]

6. Finally, divide both sides by \(30\): \[ x = \frac{60}{30} = 2 \]

Now that we've calculated \(x\), we can find the number of miles at that point:

Substituting \(x = 2\) into the equations for Dylan and Nora:

For Dylan: \[ y = 60 + 10(2) = 60 + 20 = 80 \]

For Nora: \[ y = 120 - 20(2) = 120 - 40 = 80 \]

Thus, both Dylan and Nora bike the same number of miles, which confirms that the point of intersection is indeed \((2, 80)\).