Question

1. (4 points each) solve each system by substitution or elimination:

2x + 3y = -6
-x - 3y = 12
-6x - 2y = 2
8x - 7y = -22

Explanation:

System 1: $2x + 3y = -6$; $-x - 3y = 12$

Step1: Eliminate $y$ by adding equations

Add the two equations together:
$$(2x + 3y) + (-x - 3y) = -6 + 12$$
$$x = 6$$

Step2: Substitute $x=6$ into first equation

Plug $x=6$ into $2x + 3y = -6$:
$$2(6) + 3y = -6$$
$$12 + 3y = -6$$
$$3y = -6 - 12$$
$$3y = -18$$
$$y = -6$$

System 2: $-6x - 2y = 2$; $8x - 7y = -22$

Step1: Simplify first equation for $y$

Divide $-6x - 2y = 2$ by $-2$:
$$3x + y = -1$$
$$y = -1 - 3x$$

Step2: Substitute $y=-1-3x$ into second equation

Plug $y=-1-3x$ into $8x - 7y = -22$:
$$8x - 7(-1 - 3x) = -22$$
$$8x + 7 + 21x = -22$$
$$29x + 7 = -22$$
$$29x = -22 - 7$$
$$29x = -29$$
$$x = -1$$

Step3: Solve for $y$ using $x=-1$

Substitute $x=-1$ into $y = -1 - 3x$:
$$y = -1 - 3(-1)$$
$$y = -1 + 3$$
$$y = 2$$

Answer:

1. For the system $2x + 3y = -6$ and $-x - 3y = 12$: $x=6$, $y=-6$
2. For the system $-6x - 2y = 2$ and $8x - 7y = -22$: $x=-1$, $y=2$