Linear Equations Homework 3: Graphing Linear Equations by Slope-Intercept Form Bell: Date: This is a 2-page document! Directions: Graph the following linear equations: Convert to slope-intercept form when necessary: 3x – 3y = 7 + 5 V = t + 7. y = 7 – x 8. y = -4 + 5 9. y = -5 – 3x Gina Wilson (All Things Algebra, LLC), 2012-2017
The Correct Answer and Explanation is:
Let’s break down each equation step by step and graph them based on the slope-intercept form, y=mx+by = mx + by=mx+b, where mmm is the slope and bbb is the y-intercept.
1. Equation: 3x – 3y = 7 + 5
First, simplify the right-hand side:3x−3y=123x – 3y = 123x−3y=12
Next, solve for yyy to put it into slope-intercept form:−3y=−3x+12-3y = -3x + 12−3y=−3x+12
Now, divide through by -3:y=x−4y = x – 4y=x−4
This is now in slope-intercept form y=x−4y = x – 4y=x−4, where the slope m=1m = 1m=1 and the y-intercept b=−4b = -4b=−4.
- Slope = 1 (this means the line rises 1 unit up for every 1 unit it moves to the right)
- y-intercept = -4 (the line crosses the y-axis at y=−4y = -4y=−4)
2. Equation: V = t + 7
This equation is already in slope-intercept form:V=t+7V = t + 7V=t+7
Here, the slope is 1, and the y-intercept is 7.
- Slope = 1
- y-intercept = 7
3. Equation: y = 7 – x
Rearrange the terms to match the slope-intercept form:y=−x+7y = -x + 7y=−x+7
This is in slope-intercept form, where:
- Slope = -1 (this means the line falls 1 unit for every 1 unit it moves to the right)
- y-intercept = 7
4. Equation: y = -4 + 5
Simplify the right-hand side:y=1y = 1y=1
This represents a horizontal line at y=1y = 1y=1, with:
- Slope = 0 (horizontal lines have a slope of 0)
- y-intercept = 1
5. Equation: y = -5 – 3x
Rearrange the terms:y=−3x−5y = -3x – 5y=−3x−5
This is in slope-intercept form, where:
- Slope = -3 (the line falls 3 units for every 1 unit it moves to the right)
- y-intercept = -5
Explanation:
The slope-intercept form of a line, y=mx+by = mx + by=mx+b, is very useful in graphing because it provides both the slope and the y-intercept directly. The slope (m) indicates the steepness of the line, while the y-intercept (b) is where the line crosses the y-axis.
To graph these equations:
- Start at the y-intercept point (where the line crosses the y-axis).
- Use the slope to determine the next points. If the slope is positive, the line rises as you move to the right; if it is negative, the line falls.
For example, for the equation y=x−4y = x – 4y=x−4, start at (0, -4) on the y-axis. From there, move 1 unit up and 1 unit right to plot another point at (1, -3), and so on. Repeat for all the equations to graph them accurately.
