Question

-x + y = 5
x - 5y = -9

x + y = 12
x - y = 2

2x + 5y = -24
3x - 5y = 14

Explanation:

First System of Equations: $-x + y = 5$ and $x - 5y = -9$

Step1: Add the two equations

Adding $-x + y = 5$ and $x - 5y = -9$ to eliminate $x$:
$$(-x + y) + (x - 5y) = 5 + (-9)$$
$$-x + y + x - 5y = -4$$
$$-4y = -4$$

Step2: Solve for $y$

Divide both sides by $-4$:
$$y = \frac{-4}{-4} = 1$$

Step3: Substitute $y = 1$ into $-x + y = 5$

$$-x + 1 = 5$$
Subtract 1 from both sides:
$$-x = 5 - 1 = 4$$
Multiply both sides by $-1$:
$$x = -4$$

Step1: Add the two equations

Adding $x + y = 12$ and $x - y = 2$ to eliminate $y$:
$$(x + y) + (x - y) = 12 + 2$$
$$x + y + x - y = 14$$
$$2x = 14$$

Step2: Solve for $x$

Divide both sides by 2:
$$x = \frac{14}{2} = 7$$

Step3: Substitute $x = 7$ into $x + y = 12$

$$7 + y = 12$$
Subtract 7 from both sides:
$$y = 12 - 7 = 5$$

Step1: Add the two equations

Adding $2x + 5y = -24$ and $3x - 5y = 14$ to eliminate $y$:
$$(2x + 5y) + (3x - 5y) = -24 + 14$$
$$2x + 5y + 3x - 5y = -10$$
$$5x = -10$$

Step2: Solve for $x$

Divide both sides by 5:
$$x = \frac{-10}{5} = -2$$

Step3: Substitute $x = -2$ into $2x + 5y = -24$

$$2(-2) + 5y = -24$$
$$-4 + 5y = -24$$
Add 4 to both sides:
$$5y = -24 + 4 = -20$$
Divide both sides by 5:
$$y = \frac{-20}{5} = -4$$

Answer:

$x = -4$, $y = 1$

Second System of Equations: $x + y = 12$ and $x - y = 2$