QuestionSolve The Following Systems Of Equations By Substitution:
1. $y=x+5$ $4 x+y=20$
2. $\mathrm{y}-4 \mathrm{x}=3$ $2 x-3 y=21$ 3. $\begin{array}{c} x+2 y=-1 \\ 4 x-4 y=20 \end{array}$
Word Problem
Cathy wants to buy a gym membership. One gym has a $\$ 150$ joining fee and costs $\$ 35$ per month. Another gym has no joining fee and costs $\$ 60$ per month. a. In how many months will both gym memberships cost the same? What will that cost be? b. If Cathy plans to cancel in 5 months, which is the better option for her?

Studdy Solution

STEP 1

1. We need to solve three systems of linear equations using the substitution method.
2. For the word problem, we need to determine when the cost of two gym memberships will be the same and which option is better for a 5-month duration.

STEP 2

1. Solve System of Equations 1 by Substitution.
2. Solve System of Equations 2 by Substitution.
3. Solve System of Equations 3 by Substitution.
4. Solve the Word Problem Part (a).
5. Solve the Word Problem Part (b).

STEP 3

Solve System of Equations 1 by Substitution.
Given:
1. $y = x + 5$
2. $4x + y = 20$
Substitute $y = x + 5$ into the second equation:
$4x + (x + 5) = 20$
Simplify and solve for $x$ :
$4x + x + 5 = 20$ $5x + 5 = 20$ $5x = 15$ $x = 3$
Substitute $x = 3$ back into $y = x + 5$ :
$y = 3 + 5 = 8$
Solution: $(x, y) = (3, 8)$

STEP 4

Solve System of Equations 2 by Substitution.
Given:
1. $y - 4x = 3$
2. $2x - 3y = 21$
Solve the first equation for $y$ :
$y = 4x + 3$
Substitute $y = 4x + 3$ into the second equation:
$2x - 3(4x + 3) = 21$
Simplify and solve for $x$ :
$2x - 12x - 9 = 21$ $-10x - 9 = 21$ $-10x = 30$ $x = -3$
Substitute $x = -3$ back into $y = 4x + 3$ :
$y = 4(-3) + 3 = -12 + 3 = -9$
Solution: $(x, y) = (-3, -9)$

STEP 5

Solve System of Equations 3 by Substitution.
Given:
1. $x + 2y = -1$
2. $4x - 4y = 20$
Solve the first equation for $x$ :
$x = -1 - 2y$
Substitute $x = -1 - 2y$ into the second equation:
$4(-1 - 2y) - 4y = 20$
Simplify and solve for $y$ :
$-4 - 8y - 4y = 20$ $-4 - 12y = 20$ $-12y = 24$ $y = -2$
Substitute $y = -2$ back into $x = -1 - 2y$ :
$x = -1 - 2(-2) = -1 + 4 = 3$
Solution: $(x, y) = (3, -2)$

STEP 6

Solve the Word Problem Part (a).
Let $m$ be the number of months.
Cost of Gym 1: $150 + 35m$ Cost of Gym 2: $60m$
Set the costs equal to find when they are the same:
$150 + 35m = 60m$
Solve for $m$ :
$150 = 25m$ $m = 6$
Substitute $m = 6$ into either cost equation to find the cost:
Cost: $60 \times 6 = 360$
In 6 months, both gym memberships cost the same, which is $\$360$ .

STEP 7

Solve the Word Problem Part (b).
Calculate the cost for each gym for 5 months:
Gym 1: $150 + 35 \times 5 = 150 + 175 = 325$ Gym 2: $60 \times 5 = 300$
For 5 months, Gym 2 is the better option as it costs $\$300$ compared to Gym 1's $\$325$ .

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