Question

1. $5x + 4y = -30$

$3x - 9y = -18$

1. $3x - 2y = 2$

$5x - 5y = 10$

1. $2x + 8y = 6$

$-5x - 20y = -15$

1. $3 + 2x - y = 0$

$-3 - 7y = 10x$

Explanation:

Problem 18: \( \boldsymbol{5x + 4y = -30} \) and \( \boldsymbol{3x - 9y = -18} \)

Step 1: Simplify the second equation

Divide the second equation \( 3x - 9y = -18 \) by 3:
\( \frac{3x}{3} - \frac{9y}{3} = \frac{-18}{3} \)
\( x - 3y = -6 \)
Then, solve for \( x \):
\( x = 3y - 6 \)

Step 2: Substitute \( x = 3y - 6 \) into the first equation

Substitute \( x \) into \( 5x + 4y = -30 \):
\( 5(3y - 6) + 4y = -30 \)

Step 3: Expand and simplify

\( 15y - 30 + 4y = -30 \)
Combine like terms:
\( 19y - 30 = -30 \)

Step 4: Solve for \( y \)

Add 30 to both sides:
\( 19y = 0 \)
\( y = 0 \)

Step 5: Solve for \( x \)

Substitute \( y = 0 \) into \( x = 3y - 6 \):
\( x = 3(0) - 6 = -6 \)

Problem 20: \( \boldsymbol{3x - 2y = 2} \) and \( \boldsymbol{5x - 5y = 10} \)

Step 1: Simplify the second equation

Divide the second equation \( 5x - 5y = 10 \) by 5:
\( \frac{5x}{5} - \frac{5y}{5} = \frac{10}{5} \)
\( x - y = 2 \)
Solve for \( x \):
\( x = y + 2 \)

Step 2: Substitute \( x = y + 2 \) into the first equation

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

Step 3: Expand and simplify

\( 3y + 6 - 2y = 2 \)
Combine like terms:
\( y + 6 = 2 \)

Step 4: Solve for \( y \)

Subtract 6 from both sides:
\( y = 2 - 6 = -4 \)

Step 5: Solve for \( x \)

Substitute \( y = -4 \) into \( x = y + 2 \):
\( x = -4 + 2 = -2 \)

Problem 22: \( \boldsymbol{2x + 8y = 6} \) and \( \boldsymbol{-5x - 20y = -15} \)

Step 1: Simplify both equations

Simplify the first equation \( 2x + 8y = 6 \) by dividing by 2:
\( x + 4y = 3 \)

Simplify the second equation \( -5x - 20y = -15 \) by dividing by -5:
\( x + 4y = 3 \)

Step 2: Analyze the equations

Both equations simplify to \( x + 4y = 3 \), meaning they are the same line. Thus, there are infinitely many solutions (all points on \( x + 4y = 3 \)).

Problem 24: \( \boldsymbol{3 + 2x - y = 0} \) and \( \boldsymbol{-3 - 7y = 10x} \)

Answer:

s:

1. \( x = -6 \), \( y = 0 \)
2. \( x = -2 \), \( y = -4 \)
3. Infinitely many solutions (all points on \( x + 4y = 3 \))
4. \( x = -1 \), \( y = 1 \)