Question

1. $x - y = 11$

$2x + y = 19$

1. $-2x - 9y = -25$

$-4x - 9y = -23$

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

$-6x + 3y = -12$

Explanation:

Problem 3:

Step1: Add the two equations to eliminate \( y \)

The system is:
\[

$$\begin{cases} x - y = 11 \\ 2x + y = 19 \end{cases}$$

\]
Adding the two equations: \((x - y)+(2x + y)=11 + 19\)
Simplify: \(x - y+2x + y=30\) ⇒ \(3x=30\)

Step2: Solve for \( x \)

Divide both sides by 3: \(x=\frac{30}{3}=10\)

Step3: Substitute \( x = 10 \) into the first equation to find \( y \)

Substitute into \(x - y = 11\): \(10 - y = 11\)
Subtract 10 from both sides: \(-y=11 - 10 = 1\)
Multiply both sides by - 1: \(y=-1\)

Step1: Subtract the first equation from the second to eliminate \( y \)

The system is:
\[

$$\begin{cases} -2x - 9y=-25 \\ -4x - 9y=-23 \end{cases}$$

\]
Subtract the first equation from the second: \((-4x - 9y)-(-2x - 9y)=-23-(-25)\)
Simplify: \(-4x - 9y + 2x+9y=-23 + 25\) ⇒ \(-2x = 2\)

Step2: Solve for \( x \)

Divide both sides by - 2: \(x=\frac{2}{-2}=-1\)

Step3: Substitute \( x=-1 \) into the first equation to find \( y \)

Substitute into \(-2x - 9y=-25\): \(-2\times(-1)-9y=-25\)
Simplify: \(2-9y=-25\)
Subtract 2 from both sides: \(-9y=-25 - 2=-27\)
Divide both sides by - 9: \(y=\frac{-27}{-9}=3\)

Step1: Subtract the second equation from the first to eliminate \( x \)

The system is:
\[

$$\begin{cases} -6x + 6y=6 \\ -6x + 3y=-12 \end{cases}$$

\]
Subtract the second equation from the first: \((-6x + 6y)-(-6x + 3y)=6-(-12)\)
Simplify: \(-6x + 6y + 6x-3y=6 + 12\) ⇒ \(3y = 18\)

Step2: Solve for \( y \)

Divide both sides by 3: \(y=\frac{18}{3}=6\)

Step3: Substitute \( y = 6 \) into the first equation to find \( x \)

Substitute into \(-6x + 6y=6\): \(-6x+6\times6 = 6\)
Simplify: \(-6x + 36 = 6\)
Subtract 36 from both sides: \(-6x=6 - 36=-30\)
Divide both sides by - 6: \(x=\frac{-30}{-6}=5\)

Answer:

\(x = 10\), \(y=-1\)

Problem 5: