Graphing Linear Equations in Standard Form - Answer Key Worksheet

To solve the problem and explain the solution, let's go through each part of the worksheet step by step. The task involves graphing linear equations in standard form and identifying key components such as slope and intercepts.

Problem Overview

The worksheet provides several linear equations in standard form (e.g., \( ax + by = c \)). For each equation, we need to:
1. Rewrite the equation in slope-intercept form (\( y = mx + b \)).
2. Identify the slope (\( m \)) and the y-intercept (\( b \)).
3. Graph the equation using the slope and y-intercept.

Let’s solve each part systematically.

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Step-by-Step Solutions

#### Problem 9: \( x - 5y = 10 \)

1. Rewrite in slope-intercept form:
\[
x - 5y = 10
\]
Subtract \( x \) from both sides:
\[
-5y = -x + 10
\]
Divide by \(-5\):
\[
y = \frac{1}{5}x - 2
\]

2. Identify slope and y-intercept:
- Slope (\( m \)): \( \frac{1}{5} \)
- Y-intercept (\( b \)): \(-2\)

3. Graph the equation:
- Start at the y-intercept \((0, -2)\).
- Use the slope \( \frac{1}{5} \): From \((0, -2)\), move up 1 unit and right 5 units to find another point.
- Draw a straight line through these points.

#### Problem 10: \( 2x + y = 5 \)

1. Rewrite in slope-intercept form:
\[
2x + y = 5
\]
Subtract \( 2x \) from both sides:
\[
y = -2x + 5
\]

2. Identify slope and y-intercept:
- Slope (\( m \)): \(-2\)
- Y-intercept (\( b \)): \(5\)

3. Graph the equation:
- Start at the y-intercept \((0, 5)\).
- Use the slope \(-2\): From \((0, 5)\), move down 2 units and right 1 unit to find another point.
- Draw a straight line through these points.

#### Problem 11: \( x = -1 \)

1. Identify characteristics:
- This is a vertical line where \( x \) is always \(-1\).
- There is no slope (undefined slope).
- There is no y-intercept because the line never crosses the y-axis.

2. Graph the equation:
- Draw a vertical line at \( x = -1 \).

#### Problem 12: \( x + 3y = -15 \)

1. Rewrite in slope-intercept form:
\[
x + 3y = -15
\]
Subtract \( x \) from both sides:
\[
3y = -x - 15
\]
Divide by 3:
\[
y = -\frac{1}{3}x - 5
\]

2. Identify slope and y-intercept:
- Slope (\( m \)): \(-\frac{1}{3}\)
- Y-intercept (\( b \)): \(-5\)

3. Graph the equation:
- Start at the y-intercept \((0, -5)\).
- Use the slope \(-\frac{1}{3}\): From \((0, -5)\), move down 1 unit and right 3 units to find another point.
- Draw a straight line through these points.

#### Problem 13: \( 4x + y = -4 \)

1. Rewrite in slope-intercept form:
\[
4x + y = -4
\]
Subtract \( 4x \) from both sides:
\[
y = -4x - 4
\]

2. Identify slope and y-intercept:
- Slope (\( m \)): \(-4\)
- Y-intercept (\( b \)): \(-4\)

3. Graph the equation:
- Start at the y-intercept \((0, -4)\).
- Use the slope \(-4\): From \((0, -4)\), move down 4 units and right 1 unit to find another point.
- Draw a straight line through these points.

#### Problem 14: \( x - y = -3 \)

1. Rewrite in slope-intercept form:
\[
x - y = -3
\]
Add \( y \) to both sides:
\[
x + 3 = y \quad \text{or} \quad y = x + 3
\]

2. Identify slope and y-intercept:
- Slope (\( m \)): \(1\)
- Y-intercept (\( b \)): \(3\)

3. Graph the equation:
- Start at the y-intercept \((0, 3)\).
- Use the slope \(1\): From \((0, 3)\), move up 1 unit and right 1 unit to find another point.
- Draw a straight line through these points.

#### Problem 15: \( 5x - 3y = -3 \)

1. Rewrite in slope-intercept form:
\[
5x - 3y = -3
\]
Subtract \( 5x \) from both sides:
\[
-3y = -5x - 3
\]
Divide by \(-3\):
\[
y = \frac{5}{3}x + 1
\]

2. Identify slope and y-intercept:
- Slope (\( m \)): \( \frac{5}{3} \)
- Y-intercept (\( b \)): \(1\)

3. Graph the equation:
- Start at the y-intercept \((0, 1)\).
- Use the slope \( \frac{5}{3} \): From \((0, 1)\), move up 5 units and right 3 units to find another point.
- Draw a straight line through these points.

#### Problem 16: \( 7x + 2y = -8 \)

1. Rewrite in slope-intercept form:
\[
7x + 2y = -8
\]
Subtract \( 7x \) from both sides:
\[
2y = -7x - 8
\]
Divide by 2:
\[
y = -\frac{7}{2}x - 4
\]

2. Identify slope and y-intercept:
- Slope (\( m \)): \(-\frac{7}{2}\)
- Y-intercept (\( b \)): \(-4\)

3. Graph the equation:
- Start at the y-intercept \((0, -4)\).
- Use the slope \(-\frac{7}{2}\): From \((0, -4)\), move down 7 units and right 2 units to find another point.
- Draw a straight line through these points.

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Final Answer

The solutions for each problem are summarized below:

1. Problem 9: \( y = \frac{1}{5}x - 2 \)
- Slope: \( \frac{1}{5} \)
- Y-intercept: \(-2\)

2. Problem 10: \( y = -2x + 5 \)
- Slope: \(-2\)
- Y-intercept: \(5\)

3. Problem 11: \( x = -1 \)
- Slope: Undefined
- Y-intercept: None

4. Problem 12: \( y = -\frac{1}{3}x - 5 \)
- Slope: \(-\frac{1}{3}\)
- Y-intercept: \(-5\)

5. Problem 13: \( y = -4x - 4 \)
- Slope: \(-4\)
- Y-intercept: \(-4\)

6. Problem 14: \( y = x + 3 \)
- Slope: \(1\)
- Y-intercept: \(3\)

7. Problem 15: \( y = \frac{5}{3}x + 1 \)
- Slope: \( \frac{5}{3} \)
- Y-intercept: \(1\)

8. Problem 16: \( y = -\frac{7}{2}x - 4 \)
- Slope: \(-\frac{7}{2}\)
- Y-intercept: \(-4\)

\boxed{\text{See detailed steps above for complete solutions.}}