Slope Intercept Form of a Line
Key Formula: The slope-intercept form of a linear equation is:
\[ y = mx + b \]
Where:
- \( m \) = slope of the line (steepness)
- \( b \) = y-intercept (where the line crosses the y-axis)
- \( x, y \) = coordinates of any point on the line
Understanding the Components
The Slope (m)
The slope measures how steep the line is:
- \( m > 0 \): Line rises from left to right
- \( m < 0 \): Line falls from left to right
- \( m = 0 \): Horizontal line
- \( m = \text{undefined} \): Vertical line (not in slope-intercept form)
The Y-Intercept (b)
The y-intercept is the point where the line crosses the y-axis (\( x = 0 \)):
- When \( x = 0 \), \( y = b \)
- The coordinate is \( (0, b) \)
- Can be positive, negative, or zero
5 Examples with Detailed Solutions
Example 1: Identify Slope and Y-Intercept
Problem: Given the equation \( y = 3x - 5 \), identify the slope and y-intercept.
Solution:
1. The equation is already in slope-intercept form \( y = mx + b \)
2. Compare \( y = 3x - 5 \) with \( y = mx + b \)
3. Slope (\( m \)) = coefficient of \( x \) = \( 3 \)
4. Y-intercept (\( b \)) = constant term = \( -5 \)
Answer: Slope = \( 3 \), Y-intercept = \( -5 \)
The line crosses the y-axis at \( (0, -5) \) and rises 3 units for every 1 unit to the right.
Example 2: Find Equation Given Slope and Y-Intercept
Problem: Write the equation of a line with slope \( -\frac{2}{3} \) and y-intercept \( 4 \).
Solution:
1. Start with slope-intercept form: \( y = mx + b \)
2. Substitute \( m = -\frac{2}{3} \) and \( b = 4 \)
3. Equation becomes: \( y = -\frac{2}{3}x + 4 \)
Answer: \( y = -\frac{2}{3}x + 4 \)
This line falls 2 units for every 3 units to the right and crosses the y-axis at \( (0, 4) \).
Example 3: Convert from Standard Form to Slope-Intercept Form
Problem: Convert \( 2x + 3y = 12 \) to slope-intercept form and identify the slope and y-intercept.
Solution:
1. Start with the equation: \( 2x + 3y = 12 \)
2. Isolate the y-term: \( 3y = -2x + 12 \)
3. Divide all terms by 3: \( y = -\frac{2}{3}x + 4 \)
Answer: Slope = \( -\frac{2}{3} \), Y-intercept = \( 4 \), Equation: \( y = -\frac{2}{3}x + 4 \)
The equation is now in \( y = mx + b \) form, making it easy to graph.
Example 4: Find Equation from Points
Problem: A line passes through points \( (0, -2) \) and \( (3, 4) \). Find its equation in slope-intercept form.
Solution:
1. Find the slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{3 - 0} = \frac{6}{3} = 2 \)
2. The y-intercept is where \( x = 0 \). From point \( (0, -2) \), \( b = -2 \)
3. Substitute into \( y = mx + b \): \( y = 2x - 2 \)
Answer: \( y = 2x - 2 \)
Verify: When \( x = 0 \), \( y = -2 \) ✓ When \( x = 3 \), \( y = 2(3) - 2 = 4 \) ✓
Example 5: Find Equation Given Slope and a Point
Problem: Find the equation of a line with slope \( -3 \) that passes through point \( (2, 5) \).
Solution:
1. Start with point-slope form: \( y - y_1 = m(x - x_1) \)
2. Substitute \( m = -3 \), \( (x_1, y_1) = (2, 5) \): \( y - 5 = -3(x - 2) \)
3. Simplify: \( y - 5 = -3x + 6 \)
4. Solve for y: \( y = -3x + 11 \)
Answer: \( y = -3x + 11 \)
Check: When \( x = 2 \), \( y = -3(2) + 11 = 5 \) ✓
Example 6: Parallel and Perpendicular Lines
Problem: Find equations of lines parallel and perpendicular to \( y = \frac{1}{4}x - 3 \) that pass through \( (8, 1) \).
Solution:
For parallel line:
1. Parallel lines have same slope: \( m = \frac{1}{4} \)
2. Use point-slope form with \( (8, 1) \): \( y - 1 = \frac{1}{4}(x - 8) \)
3. Simplify: \( y - 1 = \frac{1}{4}x - 2 \)
4. Solve for y: \( y = \frac{1}{4}x - 1 \)
For perpendicular line:
1. Perpendicular lines have slopes that are negative reciprocals: \( m = -\frac{1}{\frac{1}{4}} = -4 \)
2. Use point-slope form with \( (8, 1) \): \( y - 1 = -4(x - 8) \)
3. Simplify: \( y - 1 = -4x + 32 \)
4. Solve for y: \( y = -4x + 33 \)
Answer: Parallel: \( y = \frac{1}{4}x - 1 \), Perpendicular: \( y = -4x + 33 \)
Common Conversions to Slope-Intercept Form
From Standard Form: \( Ax + By = C \)
Steps:
1. Subtract \( Ax \) from both sides: \( By = -Ax + C \)
2. Divide by \( B \): \( y = -\frac{A}{B}x + \frac{C}{B} \)
Example: \( 4x - 2y = 8 \) becomes \( y = 2x - 4 \)
From Point-Slope Form: \( y - y_1 = m(x - x_1) \)
Steps:
1. Distribute \( m \): \( y - y_1 = mx - mx_1 \)
2. Add \( y_1 \) to both sides: \( y = mx - mx_1 + y_1 \)
Example: \( y - 3 = 2(x - 1) \) becomes \( y = 2x + 1 \)
Special Cases in Slope-Intercept Form
Horizontal Lines
Equation: \( y = b \) (where \( b \) is constant)
Slope: \( m = 0 \)
Example: \( y = 4 \) is a horizontal line through \( (0, 4) \)
Vertical Lines
Important: Vertical lines cannot be written in slope-intercept form!
Equation: \( x = a \) (where \( a \) is constant)
Slope: undefined
Example: \( x = 3 \) is a vertical line through \( (3, 0) \)
Practice Problems
Problem Set A: Basic Identification
- Identify slope and y-intercept: \( y = -2x + 7 \)
- Identify slope and y-intercept: \( y = \frac{3}{5}x - 2 \)
- Write equation: Slope = \( 4 \), y-intercept = \( -3 \)
- Write equation: Slope = \( -\frac{1}{2} \), y-intercept = \( 0 \)
Problem Set B: Conversions
- Convert to slope-intercept form: \( 3x + y = 9 \)
- Convert to slope-intercept form: \( 5x - 2y = 10 \)
- Find equation through \( (0, -4) \) with slope \( \frac{2}{3} \)
- Find equation through points \( (0, 2) \) and \( (4, 6) \)
Problem Set C: Applications
- Find parallel line to \( y = 3x - 1 \) through \( (2, 5) \)
- Find perpendicular line to \( y = -\frac{1}{4}x + 2 \) through \( (1, 3) \)
Answers to Practice Problems
Set A Answers
1. Slope = \( -2 \), y-intercept = \( 7 \)
2. Slope = \( \frac{3}{5} \), y-intercept = \( -2 \)
3. \( y = 4x - 3 \)
4. \( y = -\frac{1}{2}x \)
Set B Answers
1. \( y = -3x + 9 \)
2. \( y = \frac{5}{2}x - 5 \)
3. \( y = \frac{2}{3}x - 4 \)
4. Slope = \( 1 \), y-intercept = \( 2 \), Equation: \( y = x + 2 \)
Set C Answers
1. \( y = 3x - 1 \)
2. \( y = 4x - 1 \)
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