Finding Slope and Y-Intercept
This lesson explores how to find the slope and y-intercept of linear equations. The slope (m) represents the rate of change, while the y-intercept (b) is where the line crosses the y-axis. Understanding these concepts is crucial for graphing linear equations and interpreting their meanings.
Outline
- Introduction to Slope and Y-Intercept
- Finding the Slope from Two Points
- Finding the Y-Intercept from the Equation
- Examples of Finding Slope and Y-Intercept
- Exercises with Solutions
- Homework
- Revision Section
- Contact Us for Tutoring
Introduction to Slope and Y-Intercept
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:
m = \frac{y_2 - y_1}{x_2 - x_1}
The y-intercept b is the value of y when x = 0.
Examples of Finding Slope and Y-Intercept
Example 1:
Find the slope and y-intercept of the line given by the equation y = 3x - 5.
Example 2:
Find the slope and y-intercept from the points \((2, 4)\) and \((6, 12)\).
Example 3:
Determine the slope and y-intercept of 2x - 4y = 8.
Example 4:
Find the slope and y-intercept for the line passing through \((1, 3)\) and \((4, 7)\).
Example 5:
Calculate the slope and y-intercept of the line -x + y = 2.
Example 6:
Find the slope and y-intercept of the equation 5x + 2y = 10.
Exercises
Exercise 1:
Find the slope and y-intercept of the line given by y = -2x + 7.
Exercise 2:
Calculate the slope and y-intercept for the equation 3x + y = 6.
Exercise 3:
Find the slope from the points \((0, -2)\) and \((5, 8)\).
Exercise 4:
Determine the slope and y-intercept for 4x - 3y = 9.
Exercise 5:
Find the slope of the line passing through \((2, 5)\) and \((4, 7)\).
Exercise 6:
Find the slope and y-intercept of -2x + 3y = 12.
Solutions
Solution 1:
For y = -2x + 7: The slope is -2 and the y-intercept is 7.
Solution 2:
For 3x + y = 6: Rewrite as y = -3x + 6. The slope is -3 and the y-intercept is 6.
Solution 3:
For points \((0, -2)\) and \((5, 8)\): Slope \( m = \frac{8 - (-2)}{5 - 0} = 2\).
Solution 4:
For 4x - 3y = 9: Rewrite as y = \frac{4}{3}x - 3. The slope is \(\frac{4}{3}\) and the y-intercept is -3.
Solution 5:
For points \((2, 5)\) and \((4, 7)\): Slope \( m = \frac{7 - 5}{4 - 2} = 1\).
Solution 6:
For -2x + 3y = 12: Rewrite as y = \frac{2}{3}x + 4. The slope is \(\frac{2}{3}\) and the y-intercept is 4.
Homework
- Find the slope and y-intercept for the following equations: y = 4x - 2, 5x - y = 10, -3x + 2y = 8.
- Plot the lines and label the slope and y-intercept for each.
- Write a brief explanation of how to find the slope from a linear equation in standard form.
Revision
Review the following concepts:
- The slope-intercept form y = mx + b and its components.
- Finding the slope from two points.
- Finding the y-intercept from various forms of equations.
Watch the Video Tutorial
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