Lesson · Lines, Functions and Graphs · Lines

Slope-Intercept Form

Solving a linear equation for y puts the slope and the y-intercept in plain sight, which makes graphing immediate and comparison between two lines a matter of looking.

Open the interactive lesson Practice

y = mx + b m = slope, b = y-intercept to get there: solve for y graph: plot (0, b), then step by the slope y = 4 m = 0 a horizontal line

The form you graph from

Core ideas

What this lesson establishes

The same line has many equations. This one is arranged so that the two numbers you need in order to draw it are simply there to be read.

The form exposes what you need

In y = mx + b, the coefficient of x is the slope and the constant is the y-intercept. No work is required to extract them — which is the entire reason for rearranging into this shape.

Getting there is one goal: isolate y

Whatever the equation looks like, the procedure is Linear Equations and Inequalities's: undo the additions, then undo the multiplication. The only new demand is finishing in exactly the shape mx + b.

Every non-vertical line has one such equation

Vertical lines are the sole exception, since they cannot be solved for y at all. That exception is not an inconvenience; it is a signal, and “What a Function Is” explains what it signals.

The material

Solving for y, and what that buys

The rearrangement, the reading, and the graphing method it enables.

Definition · Slope-intercept form

A non-vertical line written y = mx + b, where m is the slope and (0, b) is the y-intercept.

Why b is the y-intercept takes one substitution rather than memory: the y-intercept is where x = 0, and putting x = 0 into y = mx + b gives y = b. Why m is the slope takes two: at x = 0 the height is b, at x = 1 it is m + b, so a run of 1 produces a rise of m.

Rearranging into the form

The work is Linear Equations and Inequalities's, applied to an equation that keeps x around as a passenger. Move every term that is not y to the other side, then divide by whatever multiplies y — dividing the whole side, every term:

2x + 3y = 12 3y = -2x + 12 subtract 2x from both sides y = (-2x + 12)/3 divide both sides by 3 y = -(2/3)x + 4 divide EACH term m = -2/3 b = 4

The last step is where a term gets dropped. (-2x + 12)/3 is -(2/3)x + 4, not -(2/3)x + 12: the divisor applies to every term of the numerator. Checking against “Graphing a Linear Equation” settles it — the intercepts there were (0, 4) and (6, 0), and -(2/3)(6) + 4 = 0 confirms both.

Example · 4x − 2y = 10

−2y = −4x + 10, then dividing by −2 gives y = 2x − 5. Both signs change, because both terms are divided by a negative. Check with (3, 1): the original gives 12 − 2 = 10, and the new form gives 2(3) − 5 = 1.

Dividing by a negative is the reliable source of errors in this lesson. It changes the sign of every term it touches, and changing only the first is a mistake that leaves a plausible-looking equation behind. The check by substitution costs a few seconds and catches it.

Graphing from the form

Plot (0, b). Then use the slope as a set of instructions: write it as a fraction rise/run, move that run to the right and that rise up (down, if the slope is negative), and mark the point you land on. Two points, and the line is drawn.

Example · Graphing y = −(3/4)x + 2

Start at (0, 2). The slope −3/4 says: run 4 right, rise −3, meaning 3 down. That lands on (4, −1), and indeed −(3/4)(4) + 2 = −3 + 2 = −1.

A whole-number slope is a fraction too: m = 2 is 2/1, so the instruction is 1 right and 2 up. Forgetting the invisible denominator is what turns a slope of 2 into a step of 2 right and 2 up, which is a line of slope 1.

The form also makes comparison trivial, and that is what “Parallel and Perpendicular Lines” will use: two lines written this way can be checked for parallelism by looking at two coefficients, with no computation at all.

Interactive

Three forms of one line

Slope-intercept, point-slope and standard, derived separately

Slope-intercept, beside the alternatives

Each form is derived from your two points independently and then checked by substituting both points back in. Watch the slope and the y-intercept appear in y = mx + b as the two numbers you can simply read, and nowhere else so plainly.

Example

P1: x

P1: y

P2: x

P2: y

Slope—

y-intercept—

Standard form—

Every number below is recomputed from the two points as exact fractions — nothing is rounded, and nothing is stored.

Method

Putting a line into y = mx + b

Isolate y, then read the two numbers off.

Check that it can be done

If the equation has no y at all it is vertical, and this form does not exist for it. Say so rather than forcing it.

Move everything but the y-term across

Add or subtract from both sides. The x-term goes to the right; the constant that was with y goes with it.

Divide every term by the coefficient of y

Every term, not just the first. If that coefficient is negative, expect every sign on the right to change.

Read m and b, then check one point

The coefficient of x is m; the constant is b. Substitute any convenient x into both the original and the new equation and confirm the same y comes out.

Reading m and b

Score 0 / 0

Choose the answer you can justify.

4x − 2y = 10, rearranged, read and checked

Isolate y, extract the two numbers, then verify against the original equation.

4x - 2y = 10 -2y = -4x + 10 subtract 4x y = 2x - 5 divide EVERY term by -2 read off: m = 2 b = -5 y-intercept (0, -5) check (0, -5): original 4(0) - 2(-5) = 0 + 10 = 10 TRUE check x = 3: new form y = 2(3) - 5 = 1 original 4(3) - 2(1) = 12 - 2 = 10 TRUE graph: from (0, -5), slope 2 = 2/1, so 1 right and 2 up -> (1, -3)

Dividing −2y = −4x + 10 by −2 changes both signs on the right: −4x becomes +2x and +10 becomes −5. Changing one and not the other gives y = 2x + 5, which passes no substitution check at all.

The two verifications are different in kind. The first confirms the intercept is a genuine solution; the second confirms the two equations have the same solutions, which is what rearranging is supposed to preserve.

For a faded rearrangement, start −3x + 6y = 18 from the supplied line 6y = 3x + 18. Divide every term, identify m and b, make one slope step from the intercept, and verify that new point in the original equation.

Common mistakes

Dividing only the first term(−2x + 12)/3 is −(2/3)x + 4. Leaving the constant undivided is the commonest slip in the rearrangement, and it moves the line without changing its slope, so the graph looks reasonable.

Losing a sign when dividing by a negativeDividing by −2 flips the sign of every term on the other side. Flipping one of them produces an equation that is wrong at every point except where the two lines happen to cross.

Reading m off before solving for yIn 2x + 3y = 12 the slope is not 2. The coefficient of x is only the slope once y stands alone with coefficient 1; here the slope is −2/3.

Check your understanding

✓

Finish when you can convert any non-vertical line to y = mx + b and graph it from the two numbers without a table.

Given five equations in mixed arrangements, including one with a negative y-coefficient and one horizontal, you should produce correct slope-intercept forms and sketch each from (0, b) and a slope step. “Parallel and Perpendicular Lines” compares lines in this form, so fluency here is what makes parallel and perpendicular a matter of looking.

Note: This form is the one to reach for when the question is "what does the line look like" or "how do two lines compare". When the question instead hands you a point and a slope, “Point-Slope and Standard Form”'s point-slope form gets there with less algebra — the forms are tools, and part of the skill is picking one.