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Graphs and Forms of Linear Equations

A free College Algebra lesson from the “Graphs and Forms of Linear Equations” unit, with a worked example and practice problems including step-by-step solutions.

Linear equations can be written in different forms. Slope-intercept form shows slope and y-intercept, point-slope form uses a point and slope, and standard form can reveal intercepts.

What you'll learn

Why it matters: Business plans, science labs, and graphing calculators use different line forms depending on what is known first: a starting value, a point and rate, or intercepts. Choosing the form that exposes the needed feature is part of college-ready algebra.

Worked example

Problem. Write the line with slope 3 and y-intercept -2.

  1. Use y = mx + b.
  2. m = 3 and b = -2.
  3. The equation is y = 3x - 2.

Answer: y = 3x - 2

Practice problems

1. In y = -4x + 9, what is the slope?

Show solution
  1. Warm-up: First identify exactly what the question is asking: In y = -4x + 9, what is the slope?
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. m is the coefficient of x.
  4. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.

Answer: -4

2. In y = 2x - 7, what is the y-intercept?

Show solution
  1. Core Practice: First identify exactly what the question is asking: In y = 2x - 7, what is the y-intercept?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. b is the constant term.
  4. Check the result by substituting or estimating: the response should match -7 and make sense in the original problem.

Answer: -7

3. A line with slope 5 through (2, 1) can be written as...

Choices: y - 1 = 5(x - 2) · y - 2 = 5(x - 1) · y + 1 = 2(x - 5) · y = 5x + 1

Show solution
  1. Challenge: First identify exactly what the question is asking: A line with slope 5 through (2, 1) can be written as...
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Use y - y1 = m(x - x1).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y - 1 = 5(x - 2)

4. For 3x + 2y = 12, what is the y-intercept?

Show solution
  1. Intercepts: First identify exactly what the question is asking: For 3x + 2y = 12, what is the y-intercept?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Set x = 0, then solve 2y = 12.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

5. What is the slope of 2x - y = 5?

Show solution
  1. Standard Form: First identify exactly what the question is asking: What is the slope of 2x - y = 5?
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Rewrite as y = 2x - 5.
  4. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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