Question

#12: solve the system by substitution.$\begin{cases}2x - 2y = 10\\3x - 7 = yend{cases}$#13: solve the system by substitution.$\begin{cases}x = 2y - 12\\4x + 2y = 2end{cases}$substitution but with special solutions:#14 solve the system by substitution.$\begin{cases}y = 2x - 7\\-6x + 3y = -21end{cases}$#15. solve the system by substitution.$\begin{cases}4x + 4y = 4\\-y + 9 = xend{cases}$

Explanation:

Problem #12

Step1: Substitute $y=3x-7$ into first equation

$2x - 2(3x - 7) = 10$

Step2: Expand and simplify

$2x - 6x + 14 = 10$
$-4x = 10 - 14$
$-4x = -4$

Step3: Solve for $x$

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

Step4: Find $y$ using $x=1$

$y = 3(1) - 7 = 3 - 7 = -4$

Problem #13

Step1: Substitute $x=2y-12$ into second equation

$4(2y - 12) + 2y = 2$

Step2: Expand and simplify

$8y - 48 + 2y = 2$
$10y = 2 + 48$
$10y = 50$

Step3: Solve for $y$

$y = \frac{50}{10} = 5$

Step4: Find $x$ using $y=5$

$x = 2(5) - 12 = 10 - 12 = -2$

Problem #14

Step1: Substitute $y=2x-7$ into second equation

$-6x + 3(2x - 7) = -21$

Step2: Expand and simplify

$-6x + 6x - 21 = -21$
$-21 = -21$

Problem #15

Step1: Substitute $x=-y+9$ into first equation

$4(-y + 9) + 4y = 4$

Step2: Expand and simplify

$-4y + 36 + 4y = 4$
$36 = 4$

Answer:

1. For #12: $x=1$, $y=-4$
2. For #13: $x=-2$, $y=5$
3. For #14: Infinitely many solutions (all points on $y=2x-7$)
4. For #15: No solution