Question

Elena and her husband Marc both drive to work. Elena's car has a current mileage (total distance driven) of $9,000$ and she drives $22,000$ $\mathrm{miles}$ more each year. Marc's car has a current mileage of $22,000$ and he drives $22,000$ $\mathrm{miles}$ more each year. Will the mileage for the two cars ever be equal? Explain.

• A. No; The equations have different slopes, so the lines do not intersect.
• B. Yes; The equations have different $y-$intercepts, so the lines intersect.
• C. No; The equations have different slopes, so the lines intersect.
• D. No; The equations have equal slopes but different $y-$intercepts, so the lines do not intersect.

Answer 1

The correct option is D

No; The equations have equal slopes but different $y-$intercepts, so the lines do not intersect.

Determine the mileage for the two cars ever be equal.

Explanation of correct option:

Option D:

The equation of slope form is defined as follows:

$y=mx+c$, Where, $m$ is the slope of the line relative to the $x$-axis and $c$ is the intercept of the $y$-axis.
So, according to the given conditions, Elena's car has a current mileage (total distance driven) of $9,000$ and she drives $22,000$ $\mathrm{miles}$ more each year.

Then, the equation will be $y=9,000+22,000x$.

Marc's car has a current mileage of $22,000$ and he drives $22,000$ $\mathrm{miles}$ more each year.

Then, the equation will be $y=22,000+22,000x$.

These two functions have the same value of $m$, so they have the same slopes but different $y-$intercepts.

Therefore, the lines do not intersect which means the mileage of two cars will never be equal.

Therefore, the last option (D) “No; The equations have equal slopes but different $y-$intercepts, so the lines do not intersect.” is true.

An explanation for the incorrect option:

Option A :

Elena's car has a current mileage (total distance driven) of $9,000$ and she drives $22,000$ $\mathrm{miles}$ more each year.

Then, the equation will be $y=9,000+22,000x$.

Marc's car has a current mileage of $22,000$ and he drives $22,000$ $\mathrm{miles}$ more each year.

Then, the equation will be $y=22,000+22,000x$.

These two functions have the same value of $m$ coefficient, so they have the same slopes but different $y-$intercepts.

But, option (A) has a different slope.

Therefore, option (A) is incorrect.

Option B:

As the equations have different $y-$intercepts , which implies that the lines don't intersect.

Therefore, option (B) is incorrect.

Option C :

The equations have the same slopes and different $y-$intercepts, so the lines do not intersect.

Therefore, option (C) is incorrect.

Therefore, option (D) is the correct option.