Slope-Intercept Form: Read It and Write It | Grade 8 Math | LevelKite

You already know the slope of a line that is not vertical: the change in $y$ divided by the change in $x$ between any two different points on the line. Slope tells you how steep a line is, but not where it sits. Different lines can have the same slope.

To pin down one line, you need one more fact. A line that is not vertical crosses the y-axis exactly once. The value of $y$ at that crossing point is the line's y-intercept. Every point on the y-axis has $x = 0$ , so a line with y-intercept $b$ crosses at the point $(0, b)$ .

In this lesson you will read the slope and the y-intercept straight from a linear equation, which is an equation whose graph is a straight line. Then you will go the other way and write the equation from those two numbers. Start with one equation, $y = 2x + 3$ , and watch what its two numbers do.

Some points on the line y = 2x + 3

x	Substitute into 2x + 3	y
$0$	$2(0) + 3 = 0 + 3$	$3$
$1$	$2(1) + 3 = 2 + 3$	$5$
$2$	$2(2) + 3 = 4 + 3$	$7$
$3$	$2(3) + 3 = 6 + 3$	$9$

The table shows two patterns.

The starting value. When $x = 0$ , the term $2x$ becomes $2(0) = 0$ , so only the $3$ is left and $y = 3$ . The line crosses the y-axis at $(0, 3)$ , so the y-intercept is $3$ .
The steady change. Each time $x$ goes up by $1$ , the term $2x$ gains one more $2$ , so $y$ goes up by $2$ . The slope is the change in $y$ divided by the change in $x$ , so it is $\frac{2}{1} = 2$ . Any two rows agree: from $(0, 3)$ to $(3, 9)$ , the slope is $\frac{9 - 3}{3 - 0} = \frac{6}{3} = 2$ .

So in $y = 2x + 3$ , the number multiplying $x$ is the slope, and the number added to the $x$ term is the y-intercept.

Now go the other way: start with a line and build its equation. Take any line that is not vertical, with slope $m$ and y-intercept $b$ , so it passes through $(0, b)$ . Let $(x, y)$ be any other point on the line. On a line that is not vertical, two different points never share the same value of $x$ , so $x \neq 0$ .

A line that is not vertical has the same slope between any two of its points. So the slope from $(0, b)$ to $(x, y)$ equals $m$ . That is the first equation below. Its denominator, $x - 0$ , is just $x$ , and it is not $0$ , so the fraction makes sense.

Multiply both sides by $x$ to get the second equation. Then add $b$ to both sides to get the third. Each move does the same thing to both sides, so the two sides stay equal. The point $(0, b)$ fits the last equation too, because $m(0) + b = b$ . So every point on the line makes $y = mx + b$ true.

\frac{y - b}{x - 0} = m \quad\Longrightarrow\quad y - b = mx \quad\Longrightarrow\quad y = mx + b

Key concept

Slope-intercept form

An equation written as $y = mx + b$ is in slope-intercept form, and its graph is a line that is not vertical. Here $(x, y)$ is any point on that line, $x$ can be any number, and $m$ and $b$ are fixed numbers.

$m$ , the number multiplying $x$ (the coefficient of $x$ ), is the slope. When $x$ goes up by $1$ , $y$ changes by $m$ .
$b$ , the number with no $x$ attached (the constant term), is the y-intercept. When $x = 0$ , $y = m(0) + b = b$ , so the line crosses the y-axis at $(0, b)$ .

Every line that is not vertical has exactly one equation of this form. A vertical line, such as $x = 4$ , has an undefined slope, so it has none.

y = mx + b

Rewrite each equation to match y = mx + b, then read the slope and y-intercept

Equation	Matched to y = mx + b	Slope	y-intercept	What to notice
$y = 5x + 2$	$y = 5x + 2$	$5$	$2$	It already matches the form.
$y = \frac{1}{2}x - 4$	$y = \frac{1}{2}x + (-4)$	$\frac{1}{2}$	$-4$	Subtracting $4$ is the same as adding $-4$ .
$y = -x + 6$	$y = -1x + 6$	$-1$	$6$	$-x$ means $-1$ times $x$ .
$y = 3x$	$y = 3x + 0$	$3$	$0$	Adding $0$ changes nothing, so $b = 0$ .
$y = 7$	$y = 0x + 7$	$0$	$7$	$0x$ is $0$ for every $x$ , so $m = 0$ .

A straight line is not always proportional. A proportional relationship has an equation of the form $y = kx$ . That is slope-intercept form with slope $k$ and $b = 0$ , as in the table row $y = 3x$ . Its line passes through the origin, the point $(0, 0)$ , and $\frac{y}{x} = k$ at every point where $x \neq 0$ .

When $b \neq 0$ , the line misses the origin, so the relationship is linear but not proportional. On $y = 3x + 2$ , the point $(1, 5)$ gives $\frac{y}{x} = 5$ , but $(2, 8)$ gives $\frac{y}{x} = 4$ .

A vertical line such as $x = 4$ is outside this form altogether: its slope is undefined, and it is not a function (a rule giving exactly one $y$ for each $x$ ), because it pairs $x = 4$ with every $y$ .

Worked example

Read the slope and y-intercept

Find the slope and the y-intercept of the line $y = 4x - 3$ .

Check the form. The variable y is alone on the left side, so the equation is solved for y. That means you can match it to slope-intercept form.
$y = 4x - 3$
Rewrite the subtraction as adding the opposite, because the form has a plus sign in front of b. Subtracting 3 is the same as adding negative 3, so the equation still says the same thing.
$y = 4x + (-3)$
Match the parts. The number multiplying x is the slope m, and the constant term is the y-intercept b. Keep the negative sign with the 3.
$m = 4, \quad b = -3$
Check the y-intercept. Substitute 0 for x in the original equation. The result should equal b, and it does.
$y = 4(0) - 3 = 0 - 3 = -3$
Find a second point so you can check the slope. Substitute 1 for x.
$y = 4(1) - 3 = 4 - 3 = 1$
From x = 0 to x = 1, y goes from negative 3 up to 1. Divide the change in y by the change in x. The result matches m.
$\frac{1 - (-3)}{1 - 0} = \frac{4}{1} = 4$

Answer

The slope is $4$ and the y-intercept is $-3$ , so the line crosses the y-axis at $(0, -3)$ .

Now for the second skill: writing an equation. You saw that a line that is not vertical, with slope $m$ and y-intercept $b$ , has the equation $y = mx + b$ . So when you know those two numbers, put the slope in place of $m$ and the y-intercept in place of $b$ .

Keep the sign of each number. If $b$ is negative, write the result as a subtraction, because adding a negative number is the same as subtracting its opposite. For example, $y = 2x + (-5)$ becomes $y = 2x - 5$ .

A slope and a y-intercept together fit exactly one line, so they give exactly one equation of this form.

Worked example

Write an equation from a slope and y-intercept

A line has slope $-2$ and y-intercept $5$ . Write its equation in slope-intercept form.

Start with slope-intercept form. The slope is a number, so the line is not vertical and this form fits it.
$y = mx + b$
Replace m with the slope, negative 2, and replace b with the y-intercept, 5. This works because m stands for the slope and b stands for the y-intercept.
$y = -2x + 5$
Check the y-intercept by substituting 0 for x. You should get 5.
$y = -2(0) + 5 = 0 + 5 = 5$
Find a second point so you can check the slope. Substitute 1 for x.
$y = -2(1) + 5 = -2 + 5 = 3$
From x = 0 to x = 1, y goes from 5 down to 3. Divide the change in y by the change in x. The result matches the given slope.
$\frac{3 - 5}{1 - 0} = \frac{-2}{1} = -2$

Answer

The equation of the line is $y = -2x + 5$ .

Worked example

Fraction slope and negative y-intercept

A line has slope $\frac{3}{4}$ and crosses the y-axis at the point $(0, -2)$ . Write its equation in slope-intercept form.

Name the two numbers. The problem gives the slope. The crossing point has x = 0, so its y-value is the y-intercept.
$m = \frac{3}{4}, \quad b = -2$
Substitute both numbers into slope-intercept form. The slope is a number, so the line is not vertical and the form applies.
$y = \frac{3}{4}x + (-2)$
Adding negative 2 is the same as subtracting 2, so write the equation the shorter way.
$y = \frac{3}{4}x - 2$
Check with a second point. Use x = 4, a multiple of the denominator, so the arithmetic stays in whole numbers. Then divide the change in y by the change in x from the given crossing point to the new point. The result matches the given slope.
$y = \frac{3}{4}(4) - 2 = 3 - 2 = 1, \qquad \frac{1 - (-2)}{4 - 0} = \frac{3}{4}$

Answer

The equation of the line is $y = \frac{3}{4}x - 2$ .

Here is the idea to keep. For any line that is not vertical, the equation $y = mx + b$ shows two facts at once: the number multiplying $x$ is the slope, and the constant term is the y-intercept, so the line crosses the y-axis at $(0, b)$ . You can read $m$ and $b$ from an equation solved for $y$ , or substitute them into the form to write the equation. Next, you will use these same two numbers to graph a line from its equation and to write an equation from a graph.