Understanding Slope Intercept Form
Slope intercept form is a way of writing a linear equation in the form y = mx + b, where m is the slope of the line and b is the y-intercept. This form is useful for graphing lines and understanding the relationship between the slope and the y-intercept.
Key Components of Slope Intercept Form
- Slope (m): The slope of a line is a measure of how steep it is. A positive slope indicates a line that slopes upward from left to right, while a negative slope indicates a line that slopes downward from left to right.
- Y-Intercept (b): The y-intercept is the point at which the line crosses the y-axis. It is the value of y when x is equal to 0.
Writing Equations in Slope Intercept Form
To write an equation in slope intercept form, you need to know the slope and the y-intercept of the line. Here are the steps:
- Find the slope: Calculate the slope of the line using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
- Find the y-intercept: Find the point at which the line crosses the y-axis. This will give you the value of b.
- Write the equation: Write the equation in the form y = mx + b, where m is the slope and b is the y-intercept.
📝 Note: Make sure to label the slope and y-intercept correctly when writing the equation.
Example Problems
Problem 1: Write the equation of the line that passes through the points (2, 3) and (4, 5) in slope intercept form.
Solution: First, find the slope of the line using the formula m = (y2 - y1) / (x2 - x1). Plugging in the values, we get m = (5 - 3) / (4 - 2) = 1.
Next, find the y-intercept by finding the point at which the line crosses the y-axis. Since the line passes through the point (2, 3), we can use the slope to find the y-intercept. The equation of the line is y = x + b. Plugging in the values x = 2 and y = 3, we get 3 = 2 + b. Solving for b, we get b = 1.
The equation of the line in slope intercept form is y = x + 1.
Problem 2: Write the equation of the line that passes through the point (1, 2) and has a slope of -3.
Solution: The equation of the line is y = mx + b, where m is the slope and b is the y-intercept. Since the slope is -3, the equation becomes y = -3x + b.
To find the y-intercept, plug in the values x = 1 and y = 2 into the equation. We get 2 = -3(1) + b. Solving for b, we get b = 5.
The equation of the line in slope intercept form is y = -3x + 5.
Table of Slope Intercept Form Equations
| Slope (m) | Y-Intercept (b) | Equation |
|---|---|---|
| 1 | 1 | y = x + 1 |
| -3 | 5 | y = -3x + 5 |
| 2 | -2 | y = 2x - 2 |
| -1 | 3 | y = -x + 3 |
📝 Note: This table shows examples of slope intercept form equations. The slope and y-intercept can be any real number.
What is slope intercept form?
+Slope intercept form is a way of writing a linear equation in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
How do I write an equation in slope intercept form?
+To write an equation in slope intercept form, you need to know the slope and the y-intercept of the line. First, find the slope using the formula m = (y2 - y1) / (x2 - x1). Then, find the y-intercept by finding the point at which the line crosses the y-axis.
What is the purpose of slope intercept form?
+The purpose of slope intercept form is to provide a way of writing linear equations that is useful for graphing lines and understanding the relationship between the slope and the y-intercept.
In summary, slope intercept form is a useful way of writing linear equations that allows us to easily graph lines and understand the relationship between the slope and the y-intercept. By following the steps outlined above, you can write equations in slope intercept form and solve problems involving linear equations.
