Jasmine and Sean both track the number of miles they bike each week. Jasmine was already biking 60 miles per week, and started biking 20 more miles each week. Sean was already biking 90 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 Jasmine and Sean bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Jasmine
Sean
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
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120
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180
200
Time (weeks)
Number of miles
The point of intersection is (
,
).
Great job!
Now that you have found the point of intersection, (3, 120), let’s check that work algebraically. The equations for the relationship between weeks and miles for Jasmine and Sean are shown below. Since the number of miles Jasmine bikes will be equal to the number of miles Sean bikes at the solution to this system, you can set the expression for the number of miles they each bike, 60+20x and 90+10x, equal to one another. Do that now.
Jasmine
Sean
0
1
2
3
4
5
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8
9
10
0
20
40
60
80
100
120
140
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180
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Time (weeks)
Number of miles

60+20x =
90+10x
Nice!
Let’s solve the equation for x, the number of weeks when Jasmine and Sean 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?
Jasmine’s miles = Sean's miles
60+20x = 90+10x

Multiply each side by 10x

Subtract 10x from each side

Add 10x to each side

Answer 1

To get rid of the x term on the right side of the equation \( 90 + 10x \), you can subtract \( 10x \) from each side of the equation.

Starting with the equation: \[ 60 + 20x = 90 + 10x \]

Subtracting \( 10x \) from both sides gives: \[ 60 + 20x - 10x = 90 + 10x - 10x \]

This simplifies to: \[ 60 + 10x = 90 \]

Now you can proceed to solve for \( x \).

Next, to isolate \( x \), subtract 60 from both sides: \[ 10x = 90 - 60 \] \[ 10x = 30 \]

Now, divide both sides by 10: \[ x = 3 \]

So, the solution to the equation confirms that they will bike the same number of miles after \( x = 3 \) weeks.

Thus, the point of intersection you found earlier, (3, 120), is indeed correct!