Question

1. $-4x - 15y = -17$ $-x + 5y = -13$ $(8, -1)$ 8. $-x - 7y = 14$ $-4x - 14y = 28$ $(0, -2)$ 9. $-x - 9y = 15$ $-7x - 9y = -3$ $(3, -2)$ 10. $7x - 2y = 3$ $4x - 2y = -6$ $(3, 9)$

Explanation:

Let's solve each system of linear equations step by step.

Problem 7:

System:
\[

$$\begin{cases} -4x - 15y = -17 \\ -x + 5y = -13 \end{cases}$$

\]

Step 1: Solve the second equation for \(x\)

From \(-x + 5y = -13\), we get \(x = 5y + 13\).

Step 2: Substitute \(x = 5y + 13\) into the first equation

\[

$$\begin{align*} -4(5y + 13) - 15y &= -17 \\ -20y - 52 - 15y &= -17 \\ -35y - 52 &= -17 \\ -35y &= -17 + 52 \\ -35y &= 35 \\ y &= -1 \end{align*}$$

\]

Step 3: Substitute \(y = -1\) back into \(x = 5y + 13\)

\[
x = 5(-1) + 13 = -5 + 13 = 8
\]

So the solution is \((8, -1)\), which matches the given answer.

Problem 8:

System:
\[

$$\begin{cases} -x - 7y = 14 \\ -4x - 14y = 28 \end{cases}$$

\]

Step 1: Simplify the equations

From the first equation, \(x = -7y - 14\).

Step 2: Substitute \(x = -7y - 14\) into the second equation

\[

$$\begin{align*} -4(-7y - 14) - 14y &= 28 \\ 28y + 56 - 14y &= 28 \\ 14y + 56 &= 28 \\ 14y &= 28 - 56 \\ 14y &= -28 \\ y &= -2 \end{align*}$$

\]

Step 3: Substitute \(y = -2\) back into \(x = -7y - 14\)

\[
x = -7(-2) - 14 = 14 - 14 = 0
\]

So the solution is \((0, -2)\), which matches the given answer.

Problem 9:

System:
\[

$$\begin{cases} -x - 9y = 15 \\ -7x - 9y = -3 \end{cases}$$

\]

Step 1: Subtract the first equation from the second equation

\[

$$\begin{align*} (-7x - 9y) - (-x - 9y) &= -3 - 15 \\ -7x - 9y + x + 9y &= -18 \\ -6x &= -18 \\ x &= 3 \end{align*}$$

\]

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

\[

$$\begin{align*} -3 - 9y &= 15 \\ -9y &= 15 + 3 \\ -9y &= 18 \\ y &= -2 \end{align*}$$

\]

So the solution is \((3, -2)\), which matches the given answer.

Problem 10:

System:
\[

$$\begin{cases} 7x - 2y = 3 \\ 4x - 2y = -6 \end{cases}$$

\]

Step 1: Subtract the second equation from the first equation

\[

$$\begin{align*} (7x - 2y) - (4x - 2y) &= 3 - (-6) \\ 7x - 2y - 4x + 2y &= 9 \\ 3x &= 9 \\ x &= 3 \end{align*}$$

\]

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

\[

$$\begin{align*} 7(3) - 2y &= 3 \\ 21 - 2y &= 3 \\ -2y &= 3 - 21 \\ -2y &= -18 \\ y &= 9 \end{align*}$$

\]

So the solution is \((3, 9)\), which matches the given answer.

Answer:

Let's solve each system of linear equations step by step.

Problem 7:

System:
\[

$$\begin{cases} -4x - 15y = -17 \\ -x + 5y = -13 \end{cases}$$

\]

Step 1: Solve the second equation for \(x\)

From \(-x + 5y = -13\), we get \(x = 5y + 13\).

Step 2: Substitute \(x = 5y + 13\) into the first equation

\[

$$\begin{align*} -4(5y + 13) - 15y &= -17 \\ -20y - 52 - 15y &= -17 \\ -35y - 52 &= -17 \\ -35y &= -17 + 52 \\ -35y &= 35 \\ y &= -1 \end{align*}$$

\]

Step 3: Substitute \(y = -1\) back into \(x = 5y + 13\)

\[
x = 5(-1) + 13 = -5 + 13 = 8
\]

So the solution is \((8, -1)\), which matches the given answer.

Problem 8:

System:
\[

$$\begin{cases} -x - 7y = 14 \\ -4x - 14y = 28 \end{cases}$$

\]

Step 1: Simplify the equations

From the first equation, \(x = -7y - 14\).

Step 2: Substitute \(x = -7y - 14\) into the second equation

\[

$$\begin{align*} -4(-7y - 14) - 14y &= 28 \\ 28y + 56 - 14y &= 28 \\ 14y + 56 &= 28 \\ 14y &= 28 - 56 \\ 14y &= -28 \\ y &= -2 \end{align*}$$

\]

Step 3: Substitute \(y = -2\) back into \(x = -7y - 14\)

\[
x = -7(-2) - 14 = 14 - 14 = 0
\]

So the solution is \((0, -2)\), which matches the given answer.

Problem 9:

System:
\[

$$\begin{cases} -x - 9y = 15 \\ -7x - 9y = -3 \end{cases}$$

\]

Step 1: Subtract the first equation from the second equation

\[

$$\begin{align*} (-7x - 9y) - (-x - 9y) &= -3 - 15 \\ -7x - 9y + x + 9y &= -18 \\ -6x &= -18 \\ x &= 3 \end{align*}$$

\]

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

\[

$$\begin{align*} -3 - 9y &= 15 \\ -9y &= 15 + 3 \\ -9y &= 18 \\ y &= -2 \end{align*}$$

\]

So the solution is \((3, -2)\), which matches the given answer.

Problem 10:

System:
\[

$$\begin{cases} 7x - 2y = 3 \\ 4x - 2y = -6 \end{cases}$$

\]

Step 1: Subtract the second equation from the first equation

\[

$$\begin{align*} (7x - 2y) - (4x - 2y) &= 3 - (-6) \\ 7x - 2y - 4x + 2y &= 9 \\ 3x &= 9 \\ x &= 3 \end{align*}$$

\]

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

\[

$$\begin{align*} 7(3) - 2y &= 3 \\ 21 - 2y &= 3 \\ -2y &= 3 - 21 \\ -2y &= -18 \\ y &= 9 \end{align*}$$

\]

So the solution is \((3, 9)\), which matches the given answer.