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Slope and Linear Equations Study Guide
This slope and linear equations study guide explains how to understand, calculate, graph, and write equations for straight lines. You will review what slope means, how to find slope from two points or a graph, and how to use slope-intercept, point-slope, and standard form. Use the slope notes, linear equations summary, flashcards, and practice quiz below to strengthen your algebra skills and prepare for homework, quizzes, or exams.
Key takeaways
- Slope measures the steepness and direction of a line and is calculated as rise over run, or change in y divided by change in x.
- A positive slope rises from left to right, a negative slope falls from left to right, a zero slope is horizontal, and an undefined slope is vertical.
- Slope-intercept form, y = mx + b, shows the slope m and y-intercept b directly.
- Point-slope form, y – y1 = m(x – x1), is useful when you know a slope and one point on the line.
- Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals, except for horizontal and vertical special cases.
What Is Slope?
Slope describes how steep a line is and whether it moves upward or downward as you read the graph from left to right. The slope formula is m = (y2 – y1) / (x2 – x1). This means slope equals the change in y-values divided by the change in x-values. You may also hear slope described as rise over run. The rise is the vertical change, and the run is the horizontal change. For example, if a line goes up 6 units while moving right 3 units, the slope is 6/3 = 2.
Types of Slope
There are four common slope types. A positive slope rises from left to right, such as m = 3. A negative slope falls from left to right, such as m = -2. A zero slope forms a horizontal line, such as y = 4, because the y-value does not change. An undefined slope forms a vertical line, such as x = -1, because the x-value does not change and division by zero is undefined. Recognizing these types helps you quickly interpret graphs and equations.
Common Forms of Linear Equations
A linear equation represents a straight line and can be written in several forms. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Point-slope form is y – y1 = m(x – x1), where m is the slope and (x1, y1) is a point on the line. Standard form is Ax + By = C, where A, B, and C are constants and A is usually nonnegative. Each form is useful in different situations: slope-intercept form is best for graphing, point-slope form is best when given a point and slope, and standard form is common for solving systems of equations.
How to Graph a Linear Equation
To graph a line in slope-intercept form, start with the y-intercept b on the y-axis. Then use the slope m as rise over run to find another point. For example, for y = 2x – 3, the y-intercept is -3, so plot (0, -3). The slope is 2, or 2/1, so move up 2 and right 1 to plot another point. Draw a straight line through the points. If the equation is not in slope-intercept form, you can either rewrite it or find two points by choosing x-values and solving for y.
Writing Linear Equations from Information
To write a linear equation, identify what information is given. If you know the slope and y-intercept, use y = mx + b directly. If you know the slope and one point, use point-slope form first, then simplify if needed. If you know two points, find the slope using m = (y2 – y1) / (x2 – x1), then use point-slope form with either point. For example, through (2, 5) and (4, 9), the slope is (9 – 5) / (4 – 2) = 4/2 = 2. Using point-slope form with (2, 5): y – 5 = 2(x – 2), which simplifies to y = 2x + 1.
Parallel and Perpendicular Lines
Parallel lines never intersect and have the same slope. For example, y = 3x + 1 and y = 3x – 8 are parallel because both slopes are 3. Perpendicular lines intersect at right angles, and their slopes are negative reciprocals. For example, a line with slope 2 is perpendicular to a line with slope -1/2. Horizontal and vertical lines are special: a horizontal line is perpendicular to a vertical line.
Flashcards
What does slope measure?
Slope measures the steepness and direction of a line.
What is the slope formula using two points?
m = (y2 – y1) / (x2 – x1).
What does rise over run mean?
Rise over run means vertical change divided by horizontal change.
What is slope-intercept form?
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
What is point-slope form?
Point-slope form is y – y1 = m(x – x1).
What type of slope does a horizontal line have?
A horizontal line has a slope of 0.
What type of slope does a vertical line have?
A vertical line has an undefined slope.
How are slopes of parallel lines related?
Parallel lines have the same slope.
How are slopes of perpendicular lines related?
Perpendicular lines have slopes that are negative reciprocals.
In y = -4x + 7, what are the slope and y-intercept?
The slope is -4, and the y-intercept is 7.
Quiz
1. What is the slope of a line through the points (1, 2) and (4, 8)?
- A. 2
- B. 3
- C. 6
- D. 1/2
Show answer
Answer: 2
Use m = (y2 – y1) / (x2 – x1). The slope is (8 – 2) / (4 – 1) = 6/3 = 2.
2. Which equation is in slope-intercept form?
- A. y = 3x – 5
- B. 2x + 4y = 12
- C. y – 1 = 2(x + 3)
- D. x = 7
Show answer
Answer: y = 3x – 5
Slope-intercept form is y = mx + b. The equation y = 3x – 5 matches this pattern.
3. What is the slope of the line y = -5x + 9?
- A. -5
- B. 9
- C. 5
- D. -9
Show answer
Answer: -5
In y = mx + b, m is the slope. For y = -5x + 9, the slope is -5.
4. Which line has an undefined slope?
- A. x = 4
- B. y = 4
- C. y = x + 4
- D. y = -4x
Show answer
Answer: x = 4
A vertical line has an undefined slope. Equations of vertical lines are written as x = a number.
5. What is the slope of a line perpendicular to a line with slope 3?
- A. -1/3
- B. 1/3
- C. 3
- D. -3
Show answer
Answer: -1/3
Perpendicular slopes are negative reciprocals. The negative reciprocal of 3 is -1/3.
6. Which equation represents a horizontal line?
- A. y = -2
- B. x = -2
- C. y = 2x
- D. x + y = 2
Show answer
Answer: y = -2
A horizontal line has the same y-value for every point, so it is written as y = a constant.
FAQs
How do I find slope from a graph?
Choose two clear points on the line, count the vertical change between them, then count the horizontal change. Divide rise by run to get the slope.
What is the difference between slope-intercept form and point-slope form?
Slope-intercept form, y = mx + b, shows the slope and y-intercept directly. Point-slope form, y – y1 = m(x – x1), uses a slope and one known point on the line.
Why is the slope of a vertical line undefined?
A vertical line has no horizontal change, so the run is 0. Since division by zero is undefined, the slope is undefined.
How can I tell if two lines are parallel or perpendicular?
Compare their slopes. Parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals, such as 4 and -1/4.
Next step
Turn this topic into a study session with notes, flashcards, and a practice quiz built from your own class material.
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