Question

1. $6x - 3y = 15$

$7x + 4y = 10$

1. $4x - 7y = 3$

$x - 7y = -15$
$x = $____, $y = $____ solution

1. there are a total of 56 students taking the math quiz. the number of boys is eight more than the number of girls. how many girls are taking the quiz?.. boys?

elimination:
$x = $
equation 1
$y = $
equation 2

1. washing 2 cars and 3 trucks takes 130 minutes. washing 2 cars and 5 trucks takes 190 minutes. how long does it take to wash each car?... each truck?

elimination:
$x = $
equation 1
$y = $
equation 2

1. a hotel offer two activity packages. one cost $192 and includes 3 hours of horseback riding and 2 hours of parasailing. the second costs $213 which includes 2 hours of horseback riding and 3 hours of parasailing. what is the cost per hour of horseback riding?.. parasailing?

elimination:
$x = $
equation 1
$y = $
equation 2

Explanation:

Problem 10:

Step1: Eliminate x, scale equations

Multiply first eq by 7: $42x - 21y = 105$
Multiply second eq by 6: $42x + 24y = 60$

Step2: Subtract equations

$(42x - 21y) - (42x + 24y) = 105 - 60$
$-45y = 45$

Step3: Solve for y

$y = \frac{45}{-45} = -1$

Step4: Substitute y into second eq

$7x + 4(-1) = 10$
$7x - 4 = 10$
$7x = 14$
$x = 2$

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Problem 11:

Step1: Eliminate y, subtract equations

$(4x - 7y) - (x - 7y) = 3 - (-15)$
$3x = 18$

Step2: Solve for x

$x = \frac{18}{3} = 6$

Step3: Substitute x into second eq

$6 - 7y = -15$
$-7y = -21$
$y = 3$

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Problem 12:

Step1: Define variables, set equations

Let $x$ = boys, $y$ = girls.
Eq1: $x + y = 56$
Eq2: $x = y + 8$

Step2: Substitute Eq2 into Eq1

$(y + 8) + y = 56$
$2y + 8 = 56$

Step3: Solve for y

$2y = 48$
$y = 24$

Step4: Solve for x

$x = 24 + 8 = 32$

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Problem 13:

Step1: Define variables, set equations

Let $x$ = car time, $y$ = truck time.
Eq1: $2x + 3y = 130$
Eq2: $2x + 5y = 190$

Step2: Subtract Eq1 from Eq2

$(2x + 5y) - (2x + 3y) = 190 - 130$
$2y = 60$

Step3: Solve for y

$y = \frac{60}{2} = 30$

Step4: Substitute y into Eq1

$2x + 3(30) = 130$
$2x + 90 = 130$
$2x = 40$
$x = 20$

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Problem 14:

Step1: Define variables, set equations

Let $x$ = horseback cost, $y$ = parasailing cost.
Eq1: $3x + 2y = 192$
Eq2: $2x + 3y = 213$

Step2: Eliminate x, scale equations

Multiply Eq1 by 2: $6x + 4y = 384$
Multiply Eq2 by 3: $6x + 9y = 639$

Step3: Subtract scaled Eq1 from Eq2

$(6x + 9y) - (6x + 4y) = 639 - 384$
$5y = 255$

Step4: Solve for y

$y = \frac{255}{5} = 51$

Step5: Substitute y into Eq1

$3x + 2(51) = 192$
$3x + 102 = 192$
$3x = 90$
$x = 30$

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Answer:

1. $x=2$, $y=-1$
2. $x=6$, $y=3$
3. $x=32$ (boys), $y=24$ (girls)
4. $x=20$ mins (car), $y=30$ mins (truck)
5. $x=\$30$ (horseback), $y=\$51$ (parasailing)