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Pre-Algebra

Graphing Linear Equations

Lesson

Most lines on the coordinate plane are written in slope-intercept form:

y=mx+by = mx + b
  • mm is the slope — how steep the line is. Slope = rise over run = how much yy changes when xx goes up by 1.
  • bb is the y-intercept — where the line crosses the y-axis (the value of yy when x=0x = 0).

Worked example 1

y=2x+3y = 2x + 3

Slope is 22. y-intercept is 33.

Worked example 2 — plug in

Find yy when x=4x = 4 on y=3x−5y = 3x - 5.

y=3(4)−5=12−5=7y = 3(4) - 5 = 12 - 5 = 7

Worked example 3 — x-intercept

The x-intercept is where the line crosses the x-axis. Set y=0y = 0 and solve for xx:

0=2x−6  ⟹  x=30 = 2x - 6 \implies x = 3

How to graph a line by hand

  1. Plot the y-intercept first — put a point at (0,b)(0, b).
  2. Use the slope to get the next point. If m=2m = 2, go up 2 and right 1 from (0,b)(0, b).
  3. Draw the line through both points.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Slope of y=4x+1\text{Slope of } y = 4x + 1

Problem 2

y-intercept of y=4x+1\text{y-intercept of } y = 4x + 1

Problem 3

Slope of y=−2x+7\text{Slope of } y = -2x + 7

Problem 4

y-intercept of y=5x−3\text{y-intercept of } y = 5x - 3

Practice

Standard problems matching the lesson.

Problem 5

Slope of y=3x−5\text{Slope of } y = 3x - 5

Problem 6

y-intercept of y=−x+8\text{y-intercept of } y = -x + 8

Problem 7

Slope of y=−12x+4\text{Slope of } y = -\tfrac{1}{2}x + 4

Problem 8

y-intercept of y=23x−6\text{y-intercept of } y = \tfrac{2}{3}x - 6

Problem 9

On y=2x+1, y when x=3\text{On } y = 2x + 1, \ y \text{ when } x = 3

Problem 10

On y=−x+5, y when x=2\text{On } y = -x + 5, \ y \text{ when } x = 2

Problem 11

On y=3x−4, y when x=0\text{On } y = 3x - 4, \ y \text{ when } x = 0

Problem 12

On y=5x, y when x=−2\text{On } y = 5x, \ y \text{ when } x = -2

Problem 13

x-intercept of y=2x−6\text{x-intercept of } y = 2x - 6

Problem 14

x-intercept of y=−x+4\text{x-intercept of } y = -x + 4

Problem 15

x-intercept of y=3x+9\text{x-intercept of } y = 3x + 9

Problem 16

x-intercept of y=4x−12\text{x-intercept of } y = 4x - 12

Problem 17

On y=2x+5, y when x=7\text{On } y = 2x + 5, \ y \text{ when } x = 7

Problem 18

On y=10x+20, y when x=0\text{On } y = 10x + 20, \ y \text{ when } x = 0

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

On y=−3x+7, y when x=4\text{On } y = -3x + 7, \ y \text{ when } x = 4

Problem 20

x-intercept of y=−2x+10\text{x-intercept of } y = -2x + 10

Problem 21

Slope of y=−34x+1\text{Slope of } y = -\tfrac{3}{4}x + 1

Problem 22

On y=12x+3, y when x=6\text{On } y = \tfrac{1}{2}x + 3, \ y \text{ when } x = 6

Problem 23

x-intercept of y=13x−2\text{x-intercept of } y = \tfrac{1}{3}x - 2

Problem 24

x-intercept of y=−3x+90\text{x-intercept of } y = -3x + 90

Problem 25

On y=−12x+8, y when x=6\text{On } y = -\tfrac{1}{2}x + 8, \ y \text{ when } x = 6

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