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

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?
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
Excellent!
We can subtract 10x on both sides to get rid of the 10x on the right while keeping the equation balanced. Record subtracting 10x on both sides of the equation.
Ballio's miles
=
Mary Beth's miles
 
50+15x
=
60+10x
−

10x
−

10x
Nice!
Subtract 10x on both sides and rewrite the equation to represent the remaining values.
Ballio's miles
=
Mary Beth's miles
 
50+15x
=
60+10x
−
10x
−
10x
 
=
Great job!
We have all of the variables on the left. Now, keep going. What move can you make next to get rid of 50 on the left so that the x term, 5x, is alone on the left?
Ballio's miles
=
Mary Beth's miles
 
50+15x
=
60+10x
−
10x
−
10x
 
50+5x
=
60

Add 50 to both sides

Subtract 50 from both sides

Add 5x on both sides.

GPT-4o mini · Answer

To isolate the $ x $ term on the left side, you would need to **subtract 50 from both sides** of the equation. This will help get $ 5x $ alone. Let's write it down: Starting from: $$ 50 + 5x = 60 $$ We subtract 50 from both sides: $$ 50 + 5x - 50 = 60 - 50 $$ This simplifies to: $$ 5x = 10 $$ Great! Now you can solve for $ x $ by dividing both sides by 5: $$ x = \frac{10}{5} = 2 $$ So, Ballio and Mary Beth will bike the same number of miles after **2 weeks**. To find the number of miles they both bike after 2 weeks, you can plug $ x = 2 $ back into either equation. **For Ballio:** $$ 50 + 15(2) = 50 + 30 = 80 $$ **For Mary Beth:** $$ 60 + 10(2) = 60 + 20 = 80 $$ Both equations give the same result, confirming that after 2 weeks, they will both be biking **80 miles**. Therefore, the point of intersection is $ (2, 80) $.