Point-Slope and Standard Form

The Three Forms of a Linear Equation

Form	Equation	Key Information
Slope-intercept	$y = mx + b$	slope $m$, y-intercept $b$
Point-slope	$y - y_1 = m(x - x_1)$	slope $m$, point $(x_1, y_1)$
Standard	$Ax + By = C$	$x$-intercept, $y$-intercept, total

All three represent the same line — just written differently.

Point-Slope Form

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

where $m$ is the slope and $(x_1, y_1)$ is any known point on the line.

When to use it: when you know the slope and a specific point (but not necessarily the y-intercept).

Example: Write the equation of a line with slope 3 passing through $(2, 5)$.

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

You can distribute and simplify to slope-intercept form:

$y - 5 = 3x - 6$ $y = 3x - 1$

Standard Form

$Ax + By = C$

where $A$, $B$, and $C$ are integers (typically $A > 0$).

When to use it: when dealing with word problems involving totals, budgets, or combinations of two quantities.

Example: A store sells pens for $2 and notebooks for $5. A customer spends $30 total.

$2x + 5y = 30$

where $x$ is pens and $y$ is notebooks.

Finding Intercepts from Standard Form

For $Ax + By = C$:

• x-intercept: set $y = 0$ → $x = \frac{C}{A}$
• y-intercept: set $x = 0$ → $y = \frac{C}{B}$

Example: For $3x + 4y = 24$:

• x-intercept: $(8, 0)$
• y-intercept: $(0, 6)$

Converting Point-Slope to Slope-Intercept

Distribute and isolate $y$.

$y - 3 = -2(x - 4)$

$y - 3 = -2x + 8$

$y = -2x + 11$

Converting Standard Form to Slope-Intercept

Solve for $y$.

$5x + 2y = 14$

$2y = -5x + 14$

$y = -\frac{5}{2}x + 7$

So the slope is $-\frac{5}{2}$ and the y-intercept is $7$.

Converting Slope-Intercept to Standard Form

Move all variable terms to one side and clear fractions.

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

Multiply by 4: $4y = 3x - 8$

Rearrange: $-3x + 4y = -8$ or $3x - 4y = 8$

Writing an Equation Given Two Points

1. Calculate the slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
2. Use point-slope form with either point.
3. Convert to the required form.

Example: Find the equation of the line through $(1, 4)$ and $(3, 10)$.

Slope: $m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$

Point-slope: $y - 4 = 3(x - 1)$

Slope-intercept: $y = 3x + 1$

Standard form: $-3x + y = 1$ or $3x - y = -1$

Tip 1: Know When to Use Each Form

• Given slope + point → point-slope form
• Given slope + y-intercept → slope-intercept form
• Given two quantities with a total → standard form
• Need to graph → slope-intercept form

Tip 2: Match the Answer Choices

On the SAT, look at the answer choices first. If they're in standard form, write your equation in standard form rather than converting at the end.

Tip 3: The Slope Is the Same in Every Form

No matter which form you use, the slope doesn't change. Use this to quickly eliminate wrong answer choices.

Tip 4: For Standard Form, Keep A Positive

The conventional form has $A > 0$. If $A$ is negative, multiply the entire equation by $-1$.

Tip 5: Quick Slope from Standard Form

For $Ax + By = C$, the slope is $m = -\frac{A}{B}$. This shortcut saves time.