Ballio and Mary Beth both track the number of miles they bike each week. Ballio was already biking 50 miles per week, and started biking 15 more miles each week. Mary Beth was already biking 60 miles per week, and started biking 10 more 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 Ballio and Mary Beth bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Ballio
Mary Beth
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 Ballio and Mary Beth are shown below. Since the number of miles Ballio bikes will be equal to the number of miles Mary Beth bikes at the solution to this system, you can set the expression for the number of miles they each bike, 50+15x and 60+10x, equal to one another. Do that now.
Ballio
Mary Beth
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

50+15x
=
60+10x
Nice!
Let’s solve the equation for x, the number of weeks when Ballio and Mary Beth 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, 10x, while keeping the equation balanced?
Ballio’s miles = Mary Beth's miles
50+15x = 60+10x

Multiply each side by 10x

Add 10x to each side

Subtract 10x from each side

Answer 1

To eliminate the \(10x\) term from the right side of the equation while keeping the equation balanced, you can subtract \(10x\) from each side.

Here’s how the equation looks after that step:

\[ 50 + 15x - 10x = 60 + 10x - 10x \]

This simplifies to:

\[ 50 + 5x = 60 \]

Now you can continue solving for \(x\). Would you like me to help with that?