Question

kate is trying to decide between two health clubs. company a charges a one - time fee of $70 and $35 a month. company b charges a one - time fee of $10 and $40 a month.
kate wrote two equations to represent the cost of each plan, c, after m months as shown below
$c = 35m+70$
$c = 40m + 10$
when kate graphed the equations on her casio graphing calculator, as shown below, the lines intersected at the point $(12,490)$
what does this point mean in relationship to the scenario above?
after 12 months, kate will have 490 points at the club.
after 12 people buy memberships, the club will earn $490.
after 12 months, both plans will cost $490.
after kate spends $490, she will get 12 months free.

Explanation:

In the equations \(C = 35m+70\) and \(C = 40m + 10\), \(m\) represents the number of months and \(C\) represents the cost. The intersection point \((12,490)\) of the two lines in a system of linear equations (where \(m\) is the independent variable and \(C\) is the dependent variable) means that for the value of \(m = 12\) (months), the value of \(C\) (cost) is the same for both equations. Substituting \(m = 12\) into \(C=35m + 70\): \(C=35\times12+70=420 + 70=490\). Substituting \(m = 12\) into \(C = 40m+10\): \(C=40\times12 + 10=480+10 = 490\). So, it means after 12 months, both plans cost $490.

Answer:

C. After 12 months, both plans will cost $490.