Unit 3 · Lesson 3 of 5
Slope-intercept form
You'll be able to: read the slope and y-intercept from \(y = mx + b\), rewrite \(Ax + By = C\) in that form, and graph the line from \(b\) and the slope.
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.
Find the slope of the line through \((-2, 1)\) and \((2, 9)\).
\(m = \dfrac{9 - 1}{2 - (-2)} = \dfrac{8}{4} = 2\).
Solve \(\dfrac{x}{4} + \dfrac{1}{2} = 3\).
Multiply every term by the LCD, 4: \(x + 2 = 12\).
Subtract 2: \(x = 10\).
Step 2 of 5The idea
When an equation of a line has \(y\) alone on one side, it is in slope-intercept form: \(y = mx + b\). You can read two facts straight off it.
\(y = mx + b\). \(m\), the number multiplying \(x\), is the slope. \(b\), the number standing alone, is the y-intercept: the line crosses the y-axis at \((0, b)\).
Get \(y\) alone from standard form \(Ax + By = C\): move the \(x\)-term to the right side, then divide every term by \(B\).
Graph it. Plot \((0, b)\). From there, use the slope as rise over run to find a second point. Draw the line through both.
A whole-number slope like \(2\) is the same as \(\tfrac{2}{1}\): up 2, right 1. A negative slope like \(-\tfrac{1}{2}\) means down 1, right 2.
Why this shows up on the placement test: many questions give a line in one form and ask for its slope, its intercept, or its matching graph.
Step 3 of 5Worked examples
Example 1: read the slope and the y-intercept
- \(y = -3x + 5\): the slope is \(m = -3\) and \(b = 5\), so the line crosses the y-axis at \((0, 5)\).
- \(y = 4 - \tfrac{1}{2}x\): reorder to \(y = -\tfrac{1}{2}x + 4\). So \(m = -\tfrac{1}{2}\) and \(b = 4\). The minus sign belongs to the \(x\)-term.
- \(y = 7x\): this is \(y = 7x + 0\). So \(m = 7\) and \(b = 0\): the line goes through the origin.
Example 2: rewrite \(4x - 2y = 6\), then graph it
- Subtract \(4x\) from both sides: \(-2y = -4x + 6\).
- Divide every term by \(-2\): \(y = \dfrac{-4x}{-2} + \dfrac{6}{-2} = 2x - 3\). Both terms on the right get divided, and the signs change.
- So \(m = 2\) and \(b = -3\). Plot \((0, -3)\).
- Slope \(2 = \tfrac{2}{1}\): up 2, right 1 gives \((1, -1)\). Once more gives \((2, 1)\). Draw the line.
Step 4 of 5Practice
Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.
What is the slope of \(y = 9 - \tfrac{3}{4}x\)?
Reorder: \(y = -\tfrac{3}{4}x + 9\). The number multiplying \(x\) is \(-\tfrac{3}{4}\).
Answer: \(m = -\tfrac{3}{4}\).
Where does \(y = -5x - 8\) cross the y-axis? Write the point.
Here \(b = -8\), so the y-intercept is \((0, -8)\).
Check: when \(x = 0\), \(y = -5(0) - 8 = -8\).
Write \(3x + y = 4\) in slope-intercept form.
Subtract \(3x\) from both sides: \(y = -3x + 4\).
Write \(2x - 5y = 10\) in slope-intercept form.
Subtract \(2x\): \(-5y = -2x + 10\).
Divide every term by \(-5\): \(y = \tfrac{2}{5}x - 2\).
What is the slope of the line \(6x + 4y = 12\)?
Subtract \(6x\): \(4y = -6x + 12\). Divide every term by 4: \(y = -\tfrac{3}{2}x + 3\).
Answer: \(m = -\tfrac{3}{2}\).
Write the equation of the line with slope \(-\tfrac{2}{3}\) and y-intercept \((0, 4)\).
Put \(m = -\tfrac{2}{3}\) and \(b = 4\) into \(y = mx + b\).
Answer: \(y = -\tfrac{2}{3}x + 4\).
On paper, graph \(y = -\tfrac{1}{2}x + 3\) using the y-intercept and the slope. Then open the answer and compare.
\(b = 3\): plot \((0, 3)\).
\(m = -\tfrac{1}{2}\): down 1, right 2 gives \((2, 2)\), and again gives \((4, 1)\).
Draw the line through these points. It falls gently from left to right.
Which equation matches this graph?
- \(y = -2x + 3\)
- \(y = 3x - 2\)
- \(y = 3x + 2\)
- \(y = \tfrac{1}{3}x - 2\)
The line crosses the y-axis at \((0, -2)\), so \(b = -2\). That rules out (a) and (c).
From \((0, -2)\) to \((1, 1)\): rise 3, run 1, so \(m = 3\). That rules out (d), which has slope \(\tfrac{1}{3}\).
Answer: (b) \(y = 3x - 2\).
Step 5 of 5Wrap-up
You can now read \(m\) and \(b\) from \(y = mx + b\), get \(y\) alone from \(Ax + By = C\), and graph a line by plotting \((0, b)\) and stepping by the slope.
Read or watch next: OpenStax Elementary Algebra 2e, 4.5 Use the Slope-Intercept Form of an Equation of a Line (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Intro to slope-intercept form (short videos). Quick rules: Lines and systems cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "How do I graph a slope like -3/4 starting from the y-intercept?" It knows this course and can explain things another way.