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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?

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
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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.
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=

1 answer

To find the point of intersection algebraically, we can set the equations for the number of miles each person bikes equal to one another.

Jasmine's biking equation can be represented as: \[ y_J = 60 + 20x \]

Sean's biking equation can be represented as: \[ y_S = 90 + 10x \]

To find out when they bike the same number of miles, we set the two equations equal to each other: \[ 60 + 20x = 90 + 10x \]

Now, we will solve for \( x \):

  1. Subtract \( 10x \) from both sides: \[ 60 + 20x - 10x = 90 \] \[ 60 + 10x = 90 \]

  2. Subtract 60 from both sides: \[ 10x = 30 \]

  3. Divide by 10: \[ x = 3 \]

Now, to find out how many miles that is, we can substitute \( x = 3 \) back into either equation. We'll use Jasmine's equation: \[ y_J = 60 + 20(3) \] \[ y_J = 60 + 60 \] \[ y_J = 120 \]

Thus, at \( x = 3 \) weeks, both Jasmine and Sean bike 120 miles. The point of intersection is: (3, 120).

So, you can fill in the blank as follows: The point of intersection is \( (3, 120) \).