Harper and Audrina both track the number of miles they bike each week. Harper was already biking 40 miles per week, and started biking 20 more miles each week. Audrina was already biking 100 miles per week, and started biking 10 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 Harper and Audrina bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Harper
Audrina
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Time (weeks)
Number of miles
The point of intersection is (
,
).
Nice!
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 Harper and Audrina are shown below. Since the number of miles Harper bikes will be equal to the number of miles Audrina bikes at the solution to this system, you can set the expression for the number of miles they each bike, 40+20x and 100−10x, equal to one another. Do that now.
Harper
Audrina
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Time (weeks)
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=

Answer 1

To find when Harper and Audrina bike the same number of miles, we can set their expressions equal to each other.

Harper's biking equation is:
\[ H(x) = 40 + 20x \]

Audrina's biking equation is:
\[ A(x) = 100 - 10x \]

To find the point of intersection, we set the two equations equal to each other: \[ 40 + 20x = 100 - 10x \]

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

1. Add \( 10x \) to both sides: \[ 40 + 20x + 10x = 100 \] \[ 40 + 30x = 100 \]

2. Subtract \( 40 \) from both sides: \[ 30x = 100 - 40 \] \[ 30x = 60 \]

3. Divide both sides by \( 30 \): \[ x = \frac{60}{30} \] \[ x = 2 \]

Now, we can find the number of miles they both bike at \( x = 2 \):

Using Harper's equation: \[ H(2) = 40 + 20(2) = 40 + 40 = 80 \]

Using Audrina's equation: \[ A(2) = 100 - 10(2) = 100 - 20 = 80 \]

Both equations yield the same number of miles, which is 80.

Hence, the point of intersection is: \[ (2, 80) \]

So, to fill in the blank: The point of intersection is \( (2, 80) \).