Karan needs to choose between two gym plans. he can either pay a $150\$150 $150 dollar sign, 150 joining fee and a $10\$10 $10 dollar sign, 10 monthly fee, or he can pay a $50\$50 $50 dollar sign, 50 joining fee and a $30\$30 $30 dollar sign, 30 monthly fee.

let's use ccc to represent the cumulative cost of going to the gym for mmm months, and let's use mmm to represent the month where both plans cost the same total amount.

we have two unknown values, so can create and solve a system of two equations that represent this context.

hint #22 / 7

modeling the context

with one plan, karan pays a \$150$150dollar sign, 150 joining fee and a \$10$10dollar sign, 10 monthly fee. if he goes for mmm months, he pays 10m10m10, m in monthly fees plus the joining fee, so:

c=150+10mc=150+10mc, equals, 150, plus, 10, m

hint #33 / 7

with the other plan plan, karan pays a \$50$50dollar sign, 50 joining fee and a \$30$30dollar sign, 30 monthly fee. if he goes for mmm months, he pays 30m30m30, m in monthly fees plus the joining fee, so: c=50+30mc, equals, 50, plus, 30, m

hint #44 / 7

now we have the following system of equations to represent our situation:

\begin{cases} c=150+10m c=50+30m \end{cases}

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⎪

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c=150+10m

c=50+30m

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hint #55 / 7

solving for mmm by substituting

let's substitute the value of ccc from the second equation into the first equation.

\begin{aligned} \blued{c} & =150+10m (\blued{50+30m}) & =150+10m 20m & = 100 \maroond{m} & = \maroond{5} \end{aligned}

c

(50+30m)

20m

m

​  

=150+10m

=150+10m

=100

=5

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the cumulative costs are equal in month \maroond{5}5start color maroond, 5, end color maroond.

olving for ccc by substituting

we have one unknown left. let's substitute the value we got for \maroond{m}mstart color maroond, m, end color maroond back into one of the equations to solve for ccc.

\begin{aligned} \blued{c} & =150+10\maroond{m} \blued{c} & =150+10(\maroond{5}) \blued{c} & =150+50 \blued{c} & = 200 \end{aligned}

c

c

c

c

​  

=150+10m

=150+10(5)

=150+50

=200

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either plan will cost \blued{\$200}$200start color blued, dollar sign, 200, end color blued in the month when the cumulative costs are equal.

karan will pay a cumulative cost of 200$ with either plan to go to the gym for 5 months