Graphing Linear Equations
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn three practical ways to graph linear equations: using a table of values, finding intercepts, and using slope-intercept form.
The Concept
Section titled “The Concept”Graphing a linear equation means plotting all points that satisfy y = mx + b (or any form) to show the straight line.
Three common methods:
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Table of values (most reliable for any form): Pick several x-values, solve for y, plot the (x, y) points, connect with a straight line.
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Intercepts method (great for standard form Ax + By = C):
- x-intercept: set y = 0, solve for x
- y-intercept: set x = 0, solve for y
Plot (x-intercept, 0) and (0, y-intercept), draw line through them.
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Slope-intercept method (fastest if in y = mx + b): Start at y-intercept (0, b), use slope m (rise/run) to find another point, draw line.
All methods give the same line. Choose based on the equation form.
Worked Example
Section titled “Worked Example”Graph 2x + 3y = 12 (standard form).
Intercepts method:
- x-intercept: 2x + 3(0) = 12 → 2x = 12 → x = 6 → point (6, 0)
- y-intercept: 2(0) + 3y = 12 → 3y = 12 → y = 4 → point (0, 4)
Plot (6, 0) and (0, 4), draw straight line.
Table method (verify): Pick x = 0, 3, 6:
- x = 0: 2(0) + 3y = 12 → y = 4 → (0, 4)
- x = 3: 2(3) + 3y = 12 → 6 + 3y = 12 → y = 2 → (3, 2)
- x = 6: 2(6) + 3y = 12 → 12 + 3y = 12 → y = 0 → (6, 0)
All three points fall on the same line. The table confirms the intercepts.
Slope-intercept method: Convert first → 3y = −2x + 12 → y = (−2/3)x + 4
- Start at (0, 4), slope = −2/3 → down 2, right 3 → (3, 2)
- Draw line through (0, 4) and (3, 2)
Real-World Application
Section titled “Real-World Application”Graphing helps visualize relationships:
- Budget: Graph your spending equation to see when you run out of money
- Distance: Plot distance vs. time to compare speeds of two trips
- Business: Graph revenue and cost lines to find the break-even point (where they cross)
- Temperature: Plot temperature over time to see warming or cooling trends
Being able to graph from any form means you can always visualize the relationship, no matter how the equation is given.