Algebra 1 · Grades 8, 9

Slope-Intercept Form: y = mx + b

Quick answer

Slope-intercept form writes a line as y = mx + b, where m is the slope and b is the y-intercept, the value of y when x is 0. In y = 2x + 3 the line has slope 2 and crosses the y-axis at 3. To graph it, plot the intercept first, then use the slope as rise over run to step to a second point.

What you'll learn

  • Identify the slope and y-intercept from an equation in slope-intercept form
  • Graph a line directly from y = mx + b
  • Rearrange a linear equation into slope-intercept form

The form

y=mx+by = mx + b

Two letters carry all the information:

LetterNameControls
mmslopehow steep the line is, and which way it tilts
bbyy-interceptwhere the line crosses the vertical axis

So y=2x+3y = 2x + 3 describes a line with slope 22 that crosses the yy-axis at 33. You can read both facts off without drawing anything.

Why b is where the line crosses the axis

This is worth one line of algebra rather than memorising.

The yy-axis is the set of points where x=0x = 0. So to find where the line meets it, substitute x=0x = 0:

y=m(0)+b=0+b=by = m(0) + b = 0 + b = b

The mxmx term vanishes, and what remains is bb. That is the whole reason bb is the intercept — it is the only part of the equation that survives when xx is zero.

The same substitution explains what bb means in a real situation: it is the value before anything has changed. A flat fee before any usage, a bank balance before any deposits, a temperature at time zero.

Why m is the slope

Take two points one step apart: xx and x+1x + 1.

y1=mx+by2=m(x+1)+b=mx+m+by_1 = mx + b \qquad y_2 = m(x + 1) + b = mx + m + b

Subtract them:

y2−y1=my_2 - y_1 = m

Moving one across raises the line by exactly mm. That is rise over run with a run of 11, so mm is the slope, everywhere on the line.

Graphing in two moves

  1. Plot bb on the yy-axis. That is your first point, free of charge.
  2. From there, use the slope as riserun\tfrac{\text{rise}}{\text{run}} to step to a second point.
  3. Draw the line through them.

Worked examples

Common mistakes

Practice problems

  1. Identify the slope and yy-intercept of y=4x−5y = 4x - 5.

    Hint

    Compare directly with y=mx+by = mx + b, keeping the signs.

    Answer

    m=4m = 4, b=−5b = -5

    Full solution

    The coefficient of xx is 44, so m=4m = 4. The constant is −5-5, so the line crosses the yy-axis at (0,−5)(0, -5).

  2. Identify the slope and yy-intercept of y=−25x+1y = -\tfrac{2}{5}x + 1.

    Answer

    m=−25m = -\tfrac{2}{5}, b=1b = 1

    Full solution

    The slope is negative, so the line falls: down 22 for every 55 across, starting from (0,1)(0, 1).

  3. Write 3x+y=103x + y = 10 in slope-intercept form.

    Answer

    y=−3x+10y = -3x + 10

    Full solution

    Subtract 3x3x from both sides: y=−3x+10y = -3x + 10, so m=−3m = -3 and b=10b = 10.

  4. Write 4x−2y=84x - 2y = 8 in slope-intercept form.

    Hint

    Isolate the yy term first, then divide every term by its coefficient.

    Answer

    y=2x−4y = 2x - 4

    Full solution

    Subtract 4x4x: −2y=−4x+8-2y = -4x + 8. Divide every term by −2-2: y=2x−4y = 2x - 4.

    Check at x=0x = 0: y=−4y = -4, and 4(0)−2(−4)=84(0) - 2(-4) = 8 ✓

  5. What is the yy-intercept of y=6xy = 6x?

    Answer

    00

    Full solution

    There is no constant, so b=0b = 0 and the equation is y=6x+0y = 6x + 0. The line passes through the origin.

  6. A line has slope −2-2 and yy-intercept 77. Write its equation.

    Answer

    y=−2x+7y = -2x + 7

    Full solution

    Substitute m=−2m = -2 and b=7b = 7 into y=mx+by = mx + b.

  7. Starting from the intercept of y=34x−2y = \tfrac{3}{4}x - 2, give one other point on the line.

    Hint

    Plot (0,−2)(0, -2), then step up 33 and across 44.

    Answer

    (4,1)(4, 1)

    Full solution

    The intercept is (0,−2)(0, -2). Rise 33 and run 44 gives (4,1)(4, 1).

    Check: y=34(4)−2=3−2=1y = \tfrac{3}{4}(4) - 2 = 3 - 2 = 1 ✓

  8. A tank holds 200 litres and drains at 8 litres per minute. Write an equation for the volume yy after xx minutes, and find when the tank is empty.

    Hint

    Draining means the volume decreases, so the slope is negative.

    Answer

    y=−8x+200y = -8x + 200, empty after 25 minutes.

    Full solution

    The starting volume is the intercept, b=200b = 200. Losing 88 litres a minute makes the slope −8-8, so y=−8x+200y = -8x + 200.

    Empty means y=0y = 0: 0=−8x+2000 = -8x + 200, so 8x=2008x = 200 and x=25x = 25 minutes.

    Check: −8(25)+200=0-8(25) + 200 = 0 ✓

Frequently asked questions

What does b actually mean?

It is the value of y when x is 0, which is where the line crosses the vertical axis. In a real situation it is the starting amount before anything changes: a flat fee, an initial deposit, a temperature at time zero.

How do I get an equation into y = mx + b form?

Solve it for y, using the same inverse operations you use on any equation. From 2x + y = 7, subtract 2x from both sides to get y = -2x + 7, so the slope is -2 and the intercept is 7.

What if there is no visible b?

Then b is 0 and the line passes through the origin. In y = 3x the intercept is 0, because y = 3x + 0.

What to learn next

Key terms in this lesson

Slope
Slope measures how steep a line is: the change in y divided by the change in x. A slope of 2 means the line climbs 2 units for every 1 unit across.
y-intercept
The y-intercept is where a graph crosses the vertical axis — the value of y when x is 0. In y = 2x + 3 the y-intercept is 3, so the line passes through (0, 3).

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.8.F.A.3FunctionsInterpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
  • CCSS.MATH.CONTENT.HSF.IF.C.7aInterpreting FunctionsGraph linear and quadratic functions and show intercepts, maxima, and minima.
For teachersTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.