Point-Slope and Standard Form

You've learned that slope-intercept form ($y = mx + b$) is great when you know the slope and $y$-intercept. But sometimes you're given different information—like a point on the line and the slope, or just two points with neither being on the $y$-axis. That's where other forms of linear equations become useful.

Point-Slope Form

Point-slope form is designed for situations where you know the slope and any point on the line (not necessarily the $y$-intercept).

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

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

This form is particularly handy because you can write an equation immediately without having to solve for $b$ first.

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

We have $m = 3$, $x_1 = 2$, and $y_1 = 5$.

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

That's the equation in point-slope form. You could leave it like this, or convert it to slope-intercept form:

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

Example: Write the equation of a line with slope $-\frac{1}{2}$ passing through $(-4, 3)$.

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

To convert to slope-intercept form:

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

Using Point-Slope Form with Two Points

If you're given two points and neither is the $y$-intercept, point-slope form is often faster than slope-intercept form.

Example: Write the equation of the line through $(3, 7)$ and $(5, 13)$.

Step 1: Find the slope. $$m = \frac{13 - 7}{5 - 3} = \frac{6}{2} = 3$$

Step 2: Use point-slope form with either point. Let's use $(3, 7)$. $$y - 7 = 3(x - 3)$$

Step 3 (optional): Convert to slope-intercept form. $$y - 7 = 3x - 9$$ $$y = 3x - 2$$

You can verify this works for both points. When $x = 3$, $y = 3(3) - 2 = 7$. When $x = 5$, $y = 3(5) - 2 = 13$. Both correct!

Standard Form

Standard form is:

$$Ax + By = C$$

where $A$, $B$, and $C$ are integers, and typically $A$ is non-negative (though conventions vary).

Standard form is useful in certain contexts, especially when dealing with systems of equations or when you want integer coefficients. It also makes it easy to find both intercepts quickly.

Example: The equation $3x + 2y = 12$ is in standard form.

To find the $x$-intercept, set $y = 0$: $$3x = 12$$ $$x = 4$$

To find the $y$-intercept, set $x = 0$: $$2y = 12$$ $$y = 6$$

So the line crosses the $x$-axis at $(4, 0)$ and the $y$-axis at $(0, 6)$.

Converting Between Forms

You'll often need to convert from one form to another depending on what information you need.

From slope-intercept to standard form:

Example: Convert $y = 2x - 5$ to standard form.

Move the $x$ term to the left: $$-2x + y = -5$$

Multiply by -1 to make $A$ positive: $$2x - y = 5$$

This is standard form with $A = 2$, $B = -1$, $C = 5$.

From point-slope to standard form:

Example: Convert $y - 4 = 3(x - 1)$ to standard form.

Distribute: $$y - 4 = 3x - 3$$

Move all terms to one side: $$-3x + y = 1$$

Multiply by -1: $$3x - y = -1$$

From standard form to slope-intercept:

Example: Convert $4x + 2y = 10$ to slope-intercept form.

Solve for $y$: $$2y = -4x + 10$$ $$y = -2x + 5$$

Finding Equations of Parallel and Perpendicular Lines

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals.

Example: Write the equation of a line parallel to $y = 2x + 3$ that passes through $(1, 5)$.

Parallel means same slope, so $m = 2$.

Use point-slope form: $$y - 5 = 2(x - 1)$$ $$y - 5 = 2x - 2$$ $$y = 2x + 3$$

Wait, that can't be right—it's the same equation! Let me recalculate.

$$y - 5 = 2(x - 1)$$ $$y - 5 = 2x - 2$$ $$y = 2x + 3$$

Actually, I made an error. Let me redo this.

$$y - 5 = 2(x - 1)$$ $$y - 5 = 2x - 2$$ $$y = 2x + 3$$

Hmm, I keep getting the same thing. Let me check the arithmetic: $y - 5 = 2x - 2$, so $y = 2x - 2 + 5 = 2x + 3$.

Oh wait, that's actually correct! When $x = 1$, $y = 2(1) + 3 = 5$, so the point $(1, 5)$ is on the line $y = 2x + 3$. So the line through $(1, 5)$ parallel to $y = 2x + 3$ is just $y = 2x + 3$ itself.

Let me try with a different point.

Example (revised): Write the equation of a line parallel to $y = 2x + 3$ that passes through $(2, 1)$.

Slope is still 2. $$y - 1 = 2(x - 2)$$ $$y - 1 = 2x - 4$$ $$y = 2x - 3$$

Example: Write the equation of a line perpendicular to $y = \frac{3}{4}x - 1$ through $(6, 2)$.

The original slope is $\frac{3}{4}$. The perpendicular slope is the negative reciprocal: $-\frac{4}{3}$.

$$y - 2 = -\frac{4}{3}(x - 6)$$ $$y - 2 = -\frac{4}{3}x + 8$$ $$y = -\frac{4}{3}x + 10$$

When to Use Each Form

Slope-intercept ($y = mx + b$): Best when you know the slope and $y$-intercept, or when you want to graph quickly.

Point-slope ($y - y_1 = m(x - x_1)$): Best when you know the slope and any point, or when working with two points.

Standard form ($Ax + By = C$): Best when you want integer coefficients, when finding intercepts, or when solving systems of equations.

All three forms represent the same line—they're just different ways of writing the equation.

Practice Problems

Write the equation in point-slope form: slope 4, through $(2, 3)$.
Convert $y - 5 = -2(x + 1)$ to slope-intercept form.
Convert $y = 3x - 7$ to standard form.
Find the equation of the line through $(4, 1)$ and $(6, 9)$.
Write the equation of a line parallel to $y = -x + 2$ through $(3, 5)$.
Convert $2x - 3y = 9$ to slope-intercept form.

Solutions:

$y - 3 = 4(x - 2)$
$y - 5 = -2x - 2$, so $y = -2x + 3$
$-3x + y = -7$, or $3x - y = 7$
Slope: $m = \frac{9-1}{6-4} = 4$. Point-slope: $y - 1 = 4(x - 4)$. Slope-intercept: $y = 4x - 15$.
Parallel means slope is -1. $y - 5 = -1(x - 3)$, which gives $y = -x + 8$.
$-3y = -2x + 9$, so $y = \frac{2}{3}x - 3$

Easy Mistakes to Make

In point-slope form, the signs can trip you up. The form is $y - y_1$ and $x - x_1$, so if your point is $(3, -2)$, you write $y - (-2) = m(x - 3)$, which simplifies to $y + 2 = m(x - 3)$. That double negative is where errors creep in.

When converting to standard form, forgetting that $A$, $B$, and $C$ should be integers causes problems. If you end up with fractions, multiply the entire equation by whatever denominator you need to clear them out.

Mixing up parallel and perpendicular is unfortunately common. Parallel lines have identical slopes—they never meet. Perpendicular lines have negative reciprocal slopes—they form right angles where they intersect.

Not distributing properly when converting from point-slope form leads to wrong equations. When you have $y - 4 = 3(x - 1)$, you must multiply 3 by both $x$ and $-1$ to get $y - 4 = 3x - 3$.