Question

solve each system by elimination.

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

$-8x - 6y = -6$

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

$3x - y = -16$

1. $6x + 2y - 6z = 14$

$-x - 4y + 5z = -3$
$2x - 2y - 3z = 12$

Explanation:

Problem 17:

Step1: Eliminate $y$, scale first eq

Multiply $-3x - y = -6$ by 6:
$$-18x - 6y = -36$$

Step2: Subtract second eq

Subtract $-8x - 6y = -6$ from scaled eq:
$$(-18x - 6y) - (-8x - 6y) = -36 - (-6)$$
$$-10x = -30$$

Step3: Solve for $x$

$$x = \frac{-30}{-10} = 3$$

Step4: Substitute $x=3$ to find $y$

Plug $x=3$ into $-3x - y = -6$:
$$-3(3) - y = -6$$
$$-9 - y = -6$$
$$-y = 3 \implies y = -3$$

Problem 19:

Step1: Add equations to eliminate $x$

Add $-3x + 3y = 6$ and $3x - y = -16$:
$$(-3x + 3y) + (3x - y) = 6 + (-16)$$
$$2y = -10$$

Step2: Solve for $y$

$$y = \frac{-10}{2} = -5$$

Step3: Substitute $y=-5$ to find $x$

Plug $y=-5$ into $3x - y = -16$:
$$3x - (-5) = -16$$
$$3x + 5 = -16$$
$$3x = -21 \implies x = -7$$

Problem 21:

Step1: Eliminate $x$ with first/second eq

Multiply $-x - 4y + 5z = -3$ by 6:
$$-6x - 24y + 30z = -18$$
Add to $6x + 2y - 6z = 14$:
$$(6x + 2y - 6z) + (-6x - 24y + 30z) = 14 + (-18)$$
$$-22y + 24z = -4 \tag{Eq A}$$

Step2: Eliminate $x$ with second/third eq

Multiply $-x - 4y + 5z = -3$ by 2:
$$-2x - 8y + 10z = -6$$
Add to $2x - 2y - 3z = 12$:
$$(2x - 2y - 3z) + (-2x - 8y + 10z) = 12 + (-6)$$
$$-10y + 7z = 6 \tag{Eq B}$$

Step3: Solve Eq A and Eq B

Multiply Eq A by 5: $-110y + 120z = -20$
Multiply Eq B by 11: $-110y + 77z = 66$
Subtract the two:
$$(-110y + 120z) - (-110y + 77z) = -20 - 66$$
$$43z = -86 \implies z = -2$$

Step4: Substitute $z=-2$ to find $y$

Plug $z=-2$ into Eq B:
$$-10y + 7(-2) = 6$$
$$-10y -14 = 6$$
$$-10y = 20 \implies y = -2$$

Step5: Substitute $y,z$ to find $x$

Plug $y=-2,z=-2$ into $-x -4y +5z = -3$:
$$-x -4(-2) +5(-2) = -3$$
$$-x +8 -10 = -3$$
$$-x -2 = -3 \implies -x = -1 \implies x=1$$

Answer:

1. $x=3$, $y=-3$
2. $x=-7$, $y=-5$
3. $x=1$, $y=-2$, $z=-2$