Question

1. $2x - 8y = -4$

$x - 8y = 6$

1. $8x + 3y = 20$

$4x + 3y = 4$

1. $-5x - 6y = 1$

$-5x - y = 21$

1. $-2x + 2y = -12$

$-2x + 2y = -12$

1. $3x - 5y = -21$

$-6x - 6y = -6$

1. $8x - 2y = 18$

$-4x - y = -7$

1. $12x + 3y = -27$

$6x + y = -13$

1. $12x + 3y = 24$

$3x - y = 13$

1. $-90x + 10y = -23$

$54x - 6y = 12$

1. $5x + 5y = -25$

$-4x + 8y = -28$

Explanation:

Step1: Solve Problem 28 (System of Equations)

We have the system:
\[ \begin{cases} 2x - 8y = -4 \\ x - 8y = 6 \end{cases} \]
Subtract the second equation from the first equation to eliminate \(y\):
\[ (2x - 8y) - (x - 8y) = -4 - 6 \]
Simplify:
\[ 2x - 8y - x + 8y = -10 \implies x = -10 \]
Now substitute \(x = -10\) into the second equation \(x - 8y = 6\):
\[ -10 - 8y = 6 \]
Add 10 to both sides:
\[ -8y = 16 \]
Divide by -8:
\[ y = -2 \]

Step2: Solve Problem 29 (System of Equations)

We have the system:
\[ \begin{cases} 8x + 3y = 20 \\ 4x + 3y = 4 \end{cases} \]
Subtract the second equation from the first equation to eliminate \(y\):
\[ (8x + 3y) - (4x + 3y) = 20 - 4 \]
Simplify:
\[ 8x + 3y - 4x - 3y = 16 \implies 4x = 16 \implies x = 4 \]
Substitute \(x = 4\) into the second equation \(4x + 3y = 4\):
\[ 4(4) + 3y = 4 \implies 16 + 3y = 4 \]
Subtract 16 from both sides:
\[ 3y = -12 \implies y = -4 \]

Step3: Solve Problem 30 (System of Equations)

We have the system:
\[ \begin{cases} -5x - 6y = 1 \\ -5x - y = 21 \end{cases} \]
Subtract the second equation from the first equation to eliminate \(x\):
\[ (-5x - 6y) - (-5x - y) = 1 - 21 \]
Simplify:
\[ -5x - 6y + 5x + y = -20 \implies -5y = -20 \implies y = 4 \]
Substitute \(y = 4\) into the second equation \(-5x - y = 21\):
\[ -5x - 4 = 21 \]
Add 4 to both sides:
\[ -5x = 25 \implies x = -5 \]

Step4: Solve Problem 31 (System of Equations)

We have the system:
\[ \begin{cases} -2x + 2y = -12 \\ -2x + 2y = -12 \end{cases} \]
Both equations are identical, so there are infinitely many solutions. The solution is all pairs \((x, y)\) such that \(-2x + 2y = -12\), which simplifies to \(y = x - 6\).

Step5: Solve Problem 32 (System of Equations)

We have the system:
\[ \begin{cases} 3x - 5y = -21 \\ -6x - 6y = -6 \end{cases} \]
Multiply the first equation by 2:
\[ 6x - 10y = -42 \]
Add this to the second equation:
\[ (6x - 10y) + (-6x - 6y) = -42 + (-6) \]
Simplify:
\[ -16y = -48 \implies y = 3 \]
Substitute \(y = 3\) into the first equation \(3x - 5y = -21\):
\[ 3x - 5(3) = -21 \implies 3x - 15 = -21 \]
Add 15 to both sides:
\[ 3x = -6 \implies x = -2 \]

Step6: Solve Problem 33 (System of Equations)

We have the system:
\[ \begin{cases} 8x - 2y = 18 \\ -4x - y = -7 \end{cases} \]
Multiply the second equation by 2:
\[ -8x - 2y = -14 \]
Add this to the first equation:
\[ (8x - 2y) + (-8x - 2y) = 18 + (-14) \]
Simplify:
\[ -4y = 4 \implies y = -1 \]
Substitute \(y = -1\) into the second equation \(-4x - y = -7\):
\[ -4x - (-1) = -7 \implies -4x + 1 = -7 \]
Subtract 1 from both sides:
\[ -4x = -8 \implies x = 2 \]

Step7: Solve Problem 34 (System of Equations)

We have the system:
\[ \begin{cases} 12x + 3y = -27 \\ 6x + y = -13 \end{cases} \]
Multiply the second equation by 3:
\[ 18x + 3y = -39 \]
Subtract the first equation from this:
\[ (18x + 3y) - (12x + 3y) = -39 + 27 \]
Simplify:
\[ 6x = -12 \implies x = -2 \]
Substitute \(x = -2\) into the second equation \(6x + y = -13\):
\[ 6(-2) + y = -13 \implies -12 + y = -13 \]
Add 12 to both sides:
\[ y = -1 \]

Step8: Solve Problem 35 (System of Equations)

We have the system:
\[ \begin{cases} 12x + 3y = 24 \\ 3x - y = 13 \end{cases} \]
From the second equation, solve for \(y\):
\[ y = 3x - 13 \]
Substitute into the first equation:
\[ 12x + 3(3x - 13) = 24 \]
Simplify:
\[ 12x + 9x - 39 = 24 \implies 21x = 63 \implies x = 3 \]
Substitute \(x = 3\) into \(y = 3x - 13\):
\[ y = 3(3) - 13 = 9 - 13 = -4 \]

Step9: Solve Problem 36 (System of Equations)

We have the system:
\[ \begin{cases} -90x + 10y = -23 \\ 54x - 6y = 12 \end{cases} \]
Multiply the first equation by 3:
\[ -270x + 30y = -69 \]
Multiply the second equation by 5:
\[ 270x - 30y = 60 \]
Add these two equations:
\[ (-270x + 30y) + (270x - 30y) = -69 + 60 \]
Simplify:
\[ 0 = -9 \]
This is a contradiction, so there is no solution.

Step10: Solve Problem 37 (System of Equations)

We have the system:
\[ \begin{cases} 5x + 5y = -25 \\ -4x + 8y = -28 \end{cases} \]
Simplify the first equation by dividing by 5:
\[ x + y = -5 \implies x = -5 - y \]
Substitute into the second equation:
\[ -4(-5 - y) + 8y = -28 \]
Simplify:
\[ 20 + 4y + 8y = -28 \implies 12y = -48 \implies y = -4 \]
Substitute \(y = -4\) into \(x = -5 - y\):
\[ x = -5 - (-4) = -1 \]

Answers:

28) \(x = -10\), \(y = -2\)
29) \(x = 4\), \(y = -4\)
30) \(x = -5\), \(y = 4\)
31) Infinitely many solutions (\(y = x - 6\))
32) \(x = -2\), \(y = 3\)
33) \(x = 2\), \(y = -1\)
34) \(x = -2\), \(y = -1\)
35) \(x = 3\), \(y = -4\)
36) No solution
37) \(x = -1\), \(y = -4\)

Answer:

s:

1. \(x = -10\), \(y = -2\)
2. \(x = 4\), \(y = -4\)
3. \(x = -5\), \(y = 4\)
4. Infinitely many solutions (\(y = x - 6\))
5. \(x = -2\), \(y = 3\)
6. \(x = 2\), \(y = -1\)
7. \(x = -2\), \(y = -1\)
8. \(x = 3\), \(y = -4\)
9. No solution
10. \(x = -1\), \(y = -4\)