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 the information you have is different — a point on the line plus the slope, or two arbitrary points where neither is the $y$-intercept. That's where the other forms of a linear equation 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 $(2, 5)$.

Here $m = 3$, $x_1 = 2$, 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:

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

Quick check: write the point-slope equation for a line with slope $-2$ through $(3, 1)$. Show answer$y - 1 = -2(x - 3)$. Watch the signs carefully — the form is always $y - y_1 = m(x - x_1)$, so you subtract the coordinates.

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. Using $(3, 7)$: $$y - 7 = 3(x - 3)$$

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 $(2, 1)$.

Parallel means same slope, so $m = 2$. Using point-slope form with the new point:

$$y - 1 = 2(x - 2)$$ $$y - 1 = 2x - 4$$ $$y = 2x - 3$$

The new line has the same slope (2) but a different $y$-intercept ($-3$ instead of 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 same equation, each suited to different starting information.

Practice Problems

Write the equation in point-slope form: slope 4, through $(2, 3)$. Show answer$y - 3 = 4(x - 2)$
Convert $y - 5 = -2(x + 1)$ to slope-intercept form. Show answerDistribute: $y - 5 = -2x - 2$, so $y = -2x + 3$
Convert $y = 3x - 7$ to standard form. Show answerMove the $x$ term to the left: $-3x + y = -7$, then multiply by -1 to make $A$ positive: $3x - y = 7$
Find the equation of the line through $(4, 1)$ and $(6, 9)$. Show answerSlope: $m = \frac{9-1}{6-4} = 4$. Using point-slope with $(4,1)$: $y - 1 = 4(x - 4)$. Converting: $y = 4x - 15$.
Write the equation of a line parallel to $y = -x + 2$ through $(3, 5)$. Show answerParallel means same slope, so $m = -1$. Using point-slope: $y - 5 = -1(x - 3)$, which gives $y = -x + 8$.
Convert $2x - 3y = 9$ to slope-intercept form. Show answerSolve for $y$: $-3y = -2x + 9$, so $y = \frac{2}{3}x - 3$

What's Next?

A few things to watch for. The signs in point-slope form trip students up most often: the form is $y - y_1 = m(x - x_1)$, so a point with negative coordinates produces a double negative. If your point is $(3, -2)$, the form is $y - (-2) = m(x - 3)$, which simplifies to $y + 2 = m(x - 3)$. When converting to standard form, the coefficients $A$, $B$, and $C$ should be integers — if fractions appear, multiply the whole equation by a common denominator to clear them. Parallel lines have identical slopes; perpendicular lines have negative reciprocal slopes. And when distributing in point-slope form, the slope multiplies every term inside the parentheses: $y - 4 = 3(x - 1)$ becomes $y - 4 = 3x - 3$, not $y - 4 = 3x - 1$.

The next stop is linear equations, which ties together everything from slope and intercepts through all three forms.