Question

1. $-7x + 3y = 13$

$-3x - 3y = 27$

1. $x + 3y = -23$

$-x + 5y = -25$

1. $-8x - 4y = -12$

$-8x - 5y = -13$

1. $-4x - 3y = 19$

$-4x - y = 17$

1. $-2x - 9y = 22$

$9x + 3y = -24$

1. $3x - 3y = -6$

$-9x + 9y = 18$

1. $5x + 6y = 12$

$3x + y = -11$

1. $y = -3x - 14$

$y = -2x - 12$

Explanation:

Problem 5

Step1: Add the two equations

$(-7x + 3y) + (-3x - 3y) = 13 + 27$
$-10x = 40$

Step2: Solve for $x$

$x = \frac{40}{-10} = -4$

Step3: Substitute $x=-4$ into first equation

$-7(-4) + 3y = 13$
$28 + 3y = 13$

Step4: Solve for $y$

$3y = 13 - 28 = -15$
$y = \frac{-15}{3} = -5$

Problem 6

Step1: Add the two equations

$(x + 3y) + (-x + 5y) = -23 + (-25)$
$8y = -48$

Step2: Solve for $y$

$y = \frac{-48}{8} = -6$

Step3: Substitute $y=-6$ into first equation

$x + 3(-6) = -23$
$x - 18 = -23$

Step4: Solve for $x$

$x = -23 + 18 = -5$

Problem 7

Step1: Subtract second equation from first

$(-8x - 4y) - (-8x - 5y) = -12 - (-13)$
$y = 1$

Step2: Substitute $y=1$ into first equation

$-8x - 4(1) = -12$
$-8x - 4 = -12$

Step3: Solve for $x$

$-8x = -12 + 4 = -8$
$x = \frac{-8}{-8} = 1$

Problem 8

Step1: Subtract second equation from first

$(-4x - 3y) - (-4x - y) = 19 - 17$
$-2y = 2$

Step2: Solve for $y$

$y = \frac{2}{-2} = -1$

Step3: Substitute $y=-1$ into second equation

$-4x - (-1) = 17$
$-4x + 1 = 17$

Step4: Solve for $x$

$-4x = 17 - 1 = 16$
$x = \frac{16}{-4} = -4$

Problem 9

Step1: Multiply second equation by 3

$3(9x + 3y) = 3(-24)$
$27x + 9y = -72$

Step2: Add to first equation

$(-2x - 9y) + (27x + 9y) = 22 + (-72)$
$25x = -50$

Step3: Solve for $x$

$x = \frac{-50}{25} = -2$

Step4: Substitute $x=-2$ into first equation

$-2(-2) - 9y = 22$
$4 - 9y = 22$

Step5: Solve for $y$

$-9y = 22 - 4 = 18$
$y = \frac{18}{-9} = -2$

Problem 10

Step1: Multiply first equation by 3

$3(3x - 3y) = 3(-6)$
$9x - 9y = -18$

Step2: Add to second equation

$(9x - 9y) + (-9x + 9y) = -18 + 18$
$0 = 0$
This is a true statement, so there are infinitely many solutions, expressible as $y = x + 2$ (from rearranging $3x - 3y = -6$).

Problem 11

Step1: Multiply second equation by 6

$6(3x + y) = 6(-11)$
$18x + 6y = -66$

Step2: Subtract first equation from this

$(18x + 6y) - (5x + 6y) = -66 - 12$
$13x = -78$

Step3: Solve for $x$

$x = \frac{-78}{13} = -6$

Step4: Substitute $x=-6$ into second equation

$3(-6) + y = -11$
$-18 + y = -11$

Step5: Solve for $y$

$y = -11 + 18 = 7$

Problem 12

Step1: Set the two equations equal

$-3x - 14 = -2x - 12$

Step2: Solve for $x$

$-3x + 2x = -12 + 14$
$-x = 2$
$x = -2$

Step3: Substitute $x=-2$ into first equation

$y = -3(-2) - 14 = 6 - 14 = -8$

Answer:

1. $x=-4$, $y=-5$
2. $x=-5$, $y=-6$
3. $x=1$, $y=1$
4. $x=-4$, $y=-1$
5. $x=-2$, $y=-2$
6. Infinitely many solutions: $y = x + 2$
7. $x=-6$, $y=7$
8. $x=-2$, $y=-8$