Question

1. $4x - y = -20$

$x + 6y = 20$

1. $x - 6y = 7$

$-2x - 6y = 4$

1. $4x - y = 11$

$-5x + y = -15$

1. $6x - 3y = 24$

$3x + y = 7$

solve each system by elimination.
note:

1. $8x - 5y = 17$

$-8x + 5y = -21$

1. $-2x + 7y = 1$

$2x + 9y = 15$

1. $x + 10y = 26$

$-6x - 10y = -6$

1. $-4x + 8y = 28$

$4x - y = 14$

Explanation:

Let's solve problem 20: \(

$$\begin{cases} 4x - y = -20 \\ x + 6y = 20 \end{cases}$$

\)

Step 1: Eliminate \( y \)

Multiply the first equation by 6: \( 24x - 6y = -120 \)

Step 2: Add to the second equation

\( (24x - 6y) + (x + 6y) = -120 + 20 \)
\( 25x = -100 \)

Step 3: Solve for \( x \)

\( x = \frac{-100}{25} = -4 \)

Step 4: Substitute \( x = -4 \) into the second equation

\( -4 + 6y = 20 \)
\( 6y = 24 \)
\( y = 4 \)

Now problem 21: \(

$$\begin{cases} x - 6y = 7 \\ -2x - 6y = 4 \end{cases}$$

\)

Step 1: Subtract the second equation from the first

\( (x - 6y) - (-2x - 6y) = 7 - 4 \)
\( x - 6y + 2x + 6y = 3 \)
\( 3x = 3 \)

Step 2: Solve for \( x \)

\( x = 1 \)

Step 3: Substitute \( x = 1 \) into the first equation

\( 1 - 6y = 7 \)
\( -6y = 6 \)
\( y = -1 \)

Problem 22: \(

$$\begin{cases} 4x - y = 11 \\ -5x + y = -15 \end{cases}$$

\)

Step 1: Add the two equations

\( (4x - y) + (-5x + y) = 11 + (-15) \)
\( -x = -4 \)

Step 2: Solve for \( x \)

\( x = 4 \)

Step 3: Substitute \( x = 4 \) into the first equation

\( 4(4) - y = 11 \)
\( 16 - y = 11 \)
\( y = 5 \)

Problem 23: \(

$$\begin{cases} 6x - 3y = 24 \\ 3x + y = 7 \end{cases}$$

\)

Step 1: Multiply the second equation by 3

\( 9x + 3y = 21 \)

Step 2: Add to the first equation

\( (6x - 3y) + (9x + 3y) = 24 + 21 \)
\( 15x = 45 \)

Step 3: Solve for \( x \)

\( x = 3 \)

Step 4: Substitute \( x = 3 \) into the second equation

\( 3(3) + y = 7 \)
\( 9 + y = 7 \)
\( y = -2 \)

Problem 24: \(

$$\begin{cases} 8x - 5y = 17 \\ -8x + 5y = -21 \end{cases}$$

\)

Step 1: Add the two equations

\( (8x - 5y) + (-8x + 5y) = 17 + (-21) \)
\( 0 = -4 \)
This is a contradiction, so no solution.

Problem 25: \(

$$\begin{cases} -2x + 7y = 1 \\ 2x + 9y = 15 \end{cases}$$

\)

Step 1: Add the two equations

\( (-2x + 7y) + (2x + 9y) = 1 + 15 \)
\( 16y = 16 \)

Step 2: Solve for \( y \)

\( y = 1 \)

Step 3: Substitute \( y = 1 \) into the second equation

\( 2x + 9(1) = 15 \)
\( 2x = 6 \)
\( x = 3 \)

Problem 26: \(

$$\begin{cases} x + 10y = 26 \\ -6x - 10y = -6 \end{cases}$$

\)

Step 1: Add the two equations

\( (x + 10y) + (-6x - 10y) = 26 + (-6) \)
\( -5x = 20 \)

Step 2: Solve for \( x \)

\( x = -4 \)

Step 3: Substitute \( x = -4 \) into the first equation

\( -4 + 10y = 26 \)
\( 10y = 30 \)
\( y = 3 \)

Problem 27: \(

$$\begin{cases} -4x + 8y = 28 \\ 4x - y = 14 \end{cases}$$

\) (assuming the first equation is \( -4x + 8y = 28 \))

Step 1: Add the two equations

\( (-4x + 8y) + (4x - y) = 28 + 14 \)
\( 7y = 42 \)

Step 2: Solve for \( y \)

\( y = 6 \)

Step 3: Substitute \( y = 6 \) into the second equation

\( 4x - 6 = 14 \)
\( 4x = 20 \)
\( x = 5 \)

Answer:

s:

1. \( x = -4, y = 4 \)
2. \( x = 1, y = -1 \)
3. \( x = 4, y = 5 \)
4. \( x = 3, y = -2 \)
5. No solution
6. \( x = 3, y = 1 \)
7. \( x = -4, y = 3 \)
8. \( x = 5, y = 6 \)