Before you read

Answer first — see what you already know.

Which equation represents a line with slope $5$ passing through $(1, 3)$ ?

A
$y - 3 = 5 (x - 1)$
B
$y - 1 = 5 (x - 3)$
C
$y + 3 = 5 (x + 1)$
D
$y = 5 x + 3$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

The equation $y - 7 = - 2 (x + 3)$ can be rewritten in slope-intercept form as:

A
$y = - 2 x + 1$
B
$y = - 2 x + 13$
C
$y = - 2 x - 13$
D
$y = 2 x + 1$

Check your answer

Answer:

Theory

The Point-Slope Form

If you know the slope $m$ and one point $(x_1, y_1)$ on the line, you can write the equation as:

$y - y_1 = m(x - x_1)$

Line through (2, 1) with slope 3

This is often the fastest way to write an equation when you're given a point and a slope (or two points).

Why it works: it's just the slope formula rearranged:
$m = \frac{y - y_1}{x - x_1} \quad \Rightarrow \quad y - y_1 = m(x - x_1)$

Example 1

Write the equation of a line with slope $3$ through $(2, 5)$ in point-slope form.

$y - 5 = 3(x - 2)$

That's it! You can also convert to slope-intercept if needed:

$y - 5 = 3x - 6$

$y = 3x - 1$

y - 5 = 3(x - 2)

y = 3x - 1

Example 2

Write the equation of the line through $(-1, 4)$ and $(3, -2)$ .

Step 1: Find slope: $m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}$

Step 2: Use point-slope with either point. Using $(-1, 4)$ :

$y - 4 = -\frac{3}{2}(x - (-1))$

$y - 4 = -\frac{3}{2}(x + 1)$

Step 3 (optional): Convert to slope-intercept:

$y = -\frac{3}{2}x - \frac{3}{2} + 4 = -\frac{3}{2}x + \frac{5}{2}$

y - 4 = -\frac{3}{2}(x + 1)

Theory

Converting Between Forms

The SAT may give you one form and ask for another:

Form	Equation	Best for
Slope-intercept	$y = mx + b$	Reading slope and y-intercept
Point-slope	$y - y_1 = m(x - x_1)$	Writing equations quickly
Standard	$Ax + By = C$	Finding x and y intercepts

To convert point-slope $\to$ slope-intercept: distribute and simplify.
To convert slope-intercept $\to$ standard: move the $x$ -term to the left side.

Continue this lesson in the app

2 more sections including examples, practice problems, and step-by-step solutions.

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