SAT Math · Algebra · free lesson

Slope and Forms of a Line

The idea, then worked examples solved step by step, then practice questions whose numbers change every time you try them.

The topic in one minute

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Slope and Forms of a Line

The slope of the line through $(x_1, y_1)$ and $(x_2, y_2)$ :

\begin{aligned} m &= \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} \end{aligned}

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Read what is said

Slope says how steep a line is. Between two points, the change in y is the rise, the change in x is the run, and the slope is rise over run.
Through one, two, and four, eight, the rise is eight minus two, six, and the run is four minus one, three. So the slope is two: up two for every one across.
A positive slope rises from left to right, and a negative slope falls. A slope of zero is a horizontal line. A vertical line has an undefined slope, because its run is zero.
Now the trap. Through negative two, seven, and four, negative five, start both subtractions from the same point. The rise is negative twelve, the run is six, and the slope is negative two. Mix the order, and the sign comes out wrong.
Lines often come in standard form. Solve for y. For five x minus two y equals eight, the slope is five halves and the y-intercept is negative four.
Remember: slope is rise over run, both from the same point. Solve for y, and m is the slope, b the y-intercept. Now practice it with new numbers.

The idea

What slope measures

The slope of a line says how steep it is: how far the line goes up or down for each step of 1 to the right. Take any two points of the line. The change in $y$ between them is the rise. The change in $x$ is the run. The slope is the rise divided by the run.

Definition.

For the line through $(x_1, y_1)$ and $(x_2, y_2)$ , the slope $m$ is

\begin{aligned} m &= \frac{y_2 - y_1}{x_2 - x_1} \end{aligned}

Take the line through $(1, 2)$ and $(4, 8)$ :

\begin{aligned} rise &= 8 - 2 \\ &= 6 \\ run &= 4 - 1 \\ &= 3 \\ m &= \frac{rise}{run} \\ &= \frac{6}{3} \\ &= 2 \end{aligned}

The slope is $2$ : the line climbs $2$ for every 1 it moves to the right. The dashed path in the picture is the run followed by the rise.

Any two points of the same line give the same slope. That is what makes the line straight.

A positive slope: the line rises from left to right.
A negative slope: the line falls from left to right.
A slope of zero: the line is horizontal, $y = k$ .
An undefined slope: the line is vertical, $x = k$ . The run is 0, and you cannot divide by 0.

Warning.

The change in $y$ goes on top, and both subtractions start from the same point. Mixing the order gives the right size with the wrong sign.

Slope from a table or a graph

A table of a line gives you points. Pick any two columns.

$x$	0	2	4	6
$y$	7	4	1	-2

\begin{aligned} rise &= 4 - 7 \\ &= -3 \\ run &= 2 - 0 \\ &= 2 \\ m &= \frac{rise}{run} \\ &= -\frac{3}{2} \end{aligned}

On a graph, find two points where the line passes exactly through a corner of the grid, and count the rise and the run between them. Read the scale of each axis first: one square is not always 1.

Three forms of the same line

Form	Equation	What you can read from it
Slope-intercept	$y = mx + b$	the slope $m$ and the $y$ -intercept $(0, b)$
Point-slope	$y - y_1 = m(x - x_1)$	the slope $m$ and one point $(x_1, y_1)$
Standard	$Ax + By = C$	both intercepts, quickly

What $m$ and $b$ do in $y = mx + b$ : the slope turns the line. The intercept moves it up or down without turning it.

The next picture holds the $y$ -intercept at 1 and lets the slope run from $-3$ to 3. Watch the line pass through horizontal when the slope is 0.

Moving between the forms

Point-slope to slope-intercept. Start from $y - 5 = 2(x - 3)$ . Add 5 to both sides and multiply out the bracket:

So the line is $y = 2x - 1$ .

Standard to slope-intercept. Start from $3x + 4y = 12$ . Take $3x$ from both sides and divide by 4:

The number multiplying $x$ is the slope, $-\frac{3}{4}$ . The number standing alone is the $y$ -intercept, 3.

Rule.

The slope of $Ax + By = C$ is $-\frac{A}{B}$ , and its $y$ -intercept is $\frac{C}{B}$ . You get both by solving for $y$ .

Slope-intercept to standard. Bring the $x$ term to the left and multiply through to clear any fraction. From $y = -\frac{3}{4}x + 3$ , multiplying by 4 and adding $3x$ gives back $3x + 4y = 12$ .

Note.

On the SAT a question on this topic usually asks for one of four things: the slope from two points, a table or a graph; the equation of a line from a point and a slope or from two points; the slope or an intercept of a line given in standard form; or a missing coordinate when the slope is known.

Worked examples

Each example is one of the kinds of question the test asks about slope and the equation of a line. The numbers in the working are found by the engine.

Example.

1. Slope from two points. What is the slope of the line through $(-2, 7)$ and $(4, -5)$ ?

Subtract the $y$ values for the rise and the $x$ values for the run, starting from the same point both times. Here both start from $(4, -5)$ .

\begin{aligned} rise &= -5 - 7 \\ &= -12 \\ run &= 4 - (-2) \\ &= 6 \\ m &= \frac{rise}{run} \\ &= \frac{-12}{6} \\ &= -2 \end{aligned}

The slope is $-2$ . It is negative, and the picture agrees: the line falls from left to right.

Example.

2. From a table to an equation. The table shows some values of a linear relationship. Write its equation in the form $y = mx + b$ .

$x$	1	3	5	7
$y$	4	10	16	22

Any two columns give the slope. Then put one point into $y = mx + b$ to find $b$ . The first column, $(1, 4)$ , is used here.

\begin{aligned} m &= \frac{10 - 4}{3 - 1} \\ &= 3 \\ b &= 4 - 1m \\ &= 4 - 1 \times 3 \\ &= 1 \\ y &= mx + b \\ &= 3x + 1 \end{aligned}

The equation is $y = 3x + 1$ .

\begin{aligned} y &= mx + b \\ &= 3 \times 7 + 1 \\ &= 22 \end{aligned}

The check puts in $x = 7$ and gives $22$ , the last column of the table.

Example.

3. From a graph. The graph of a line is shown. What is its equation in slope-intercept form?

The line crosses the $y$ -axis at $(0, 3)$ , so $b$ is 3. For the slope, use two points on grid corners: $(-2, 4)$ and $(4, 1)$ .

\begin{aligned} rise &= 1 - 4 \\ &= -3 \\ run &= 4 - (-2) \\ &= 6 \\ m &= \frac{rise}{run} \\ &= \frac{-3}{6} \\ &= -\frac{1}{2} \\ y &= mx + b \\ &= -\frac{1}{2}x + 3 \end{aligned}

The equation is $y =$ $-\frac{1}{2}x + 3$ . The line goes down 1 for every 2 to the right.

Example.

4. Slope and intercept from standard form. What are the slope and the $y$ -intercept of the graph of $5x - 2y = 8$ ?

Solve for $y$ : take $5x$ from both sides, then divide both sides by $-2$ .

\begin{aligned} y &= \frac{8 - 5x}{-2} \\ &= -4 + \frac{5}{2}x \end{aligned}

The number multiplying $x$ is the slope and the number alone is the $y$ -intercept. The rule for $Ax + By = C$ gives the same two numbers without the rearranging:

\begin{aligned} m &= \frac{-5}{-2} \\ &= \frac{5}{2} \\ b &= \frac{8}{-2} \\ &= -4 \end{aligned}

The slope is $\frac{5}{2}$ and the line crosses the $y$ -axis at $(0, {-4})$ . Notice the sign: dividing by a negative $B$ made the slope positive.

Example.

5. The line through two points. Write the equation of the line through $(-1, 6)$ and $(3, -2)$ in slope-intercept form and in standard form.

First the slope. Then point-slope form with the point $(-1, 6)$ is $y - 6 = m(x + 1)$ , and solving it for $y$ gives the intercept.

\begin{aligned} m &= \frac{-2 - 6}{3 - (-1)} \\ &= -2 \\ b &= 6 - (-1)m \\ &= 6 - (-1) \times (-2) \\ &= 4 \\ y &= mx + b \\ &= -2x + 4 \end{aligned}

In slope-intercept form the line is $y = -2x + 4$ .

\begin{aligned} y &= mx + b \\ &= -2 \times 3 + 4 \\ &= -2 \end{aligned}

The check puts in the other point: $x = 3$ gives $-2$ , as it should. Adding ${2}x$ to both sides gives the standard form ${2}x + y = {4}$ .

Example.

6. A missing coordinate. The line through $(2, 3)$ and $(6, k)$ has a slope of $\frac{5}{2}$ . What is $k$ ?

Write the slope from the two points and set it equal to the slope you are given. The only unknown is $k$ .

\begin{aligned} \frac{k - 3}{6 - 2} &= \frac{5}{2} \\ k &= 13 \\ m &= \frac{k - 3}{6 - 2} \\ &= \frac{13 - 3}{6 - 2} \\ &= \frac{5}{2} \end{aligned}

So $k$ is $13$ , and the check gives back the slope $\frac{5}{2}$ .

Practice

Answer SAT-style questions on slope and forms of a line. Each answer is checked, with the full solution, and every try brings new numbers. Practice needs a free account: it keeps your progress.

Practise slope and forms of a line with new numbers