Point-Slope Form

Q: What point does $y + 6 = 5(x + 2)$ pass through?

$(-2, -6)$. Match the equation to the form $y - y_1 = m(x - x_1)$: rewriting $y + 6$ as $y - (-6)$ and $x + 2$ as $x - (-2)$ shows $y_1 = -6$ and $x_1 = -2$. Plus signs in the equation mean negative coordinates.

Point-slope form is the fastest way to write the equation of a line when you know one point on it and its slope. The form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the known point. No solving required — you substitute the three numbers and you're done.

Students often treat point-slope form as a stepping stone to $y = mx + b$ , but it's worth knowing on its own. Plenty of problems hand you a point that is not the $y$ -intercept, and point-slope form takes that information directly, while slope-intercept form makes you solve for $b$ first.

Where the formula comes from

Point-slope form is the slope formula in disguise. Take a known point $(x_1, y_1)$ on a line with slope $m$ , and let $(x, y)$ be any other point on the line. The slope between them must equal $m$ : $\dfrac{y - y_1}{x - x_1} = m$ .

Multiply both sides by $x - x_1$ and you get $y - y_1 = m(x - x_1)$ — point-slope form. That's the whole derivation. So when you use this form, you're really just saying: the slope from my known point to any point on the line is $m$ .

Reading the equation — watch the signs

The formula has built-in minus signs, so the numbers you see in a finished equation are not always the coordinates of the point. In $y - 3 = 2(x - 5)$ , the point is $(5, 3)$ — the signs match the formula exactly. But in $y + 4 = 2(x - 1)$ , rewrite $y + 4$ as $y - (-4)$ : the point is $(1, -4)$ , not $(1, 4)$ .

The rule: whatever number makes the parentheses read as a subtraction is the coordinate. A plus sign in the equation means the coordinate is negative. The slope has no such trap — $m$ is always the number multiplying the parentheses, sign included.

Graphing from point-slope form

Point-slope form is built for graphing: it hands you a starting point and a direction. Plot the point $(x_1, y_1)$ first. Then use the slope to step to a second point — a slope of $2$ means up $2$ units for every $1$ unit right. Connect the two points and extend the line.

The graph below shows $y - 1 = 2(x - 1)$ : the line through $(1, 1)$ with slope $2$ . Starting at $(1, 1)$ and stepping right $1$ , up $2$ lands on $(2, 3)$ , which is also on the line.

Worked examples

Example 1: write the equation from a point and a slope

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

Write the form

y - y_1 = m(x - x_1)

Identify the pieces

m = 3, \quad (x_1, y_1) = (2, 5)

Substitute

y - 5 = 3(x - 2)

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

Example 2: a negative coordinate

Write the equation of the line with slope $-2$ through $(-1, 4)$ in point-slope form.

Substitute into the form

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

Simplify the double negative

y - 4 = -2(x + 1)

Answer: $y - 4 = -2(x + 1)$

Example 3: from two points

Write the equation of the line through $(1, 2)$ and $(4, 11)$ in point-slope form.

Find the slope first

m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{11 - 2}{4 - 1}

Simplify

m = \dfrac{9}{3} = 3

Use either point — take

(1, 2)

y - 2 = 3(x - 1)

Check with the other point:

11 - 2 = 9

and

3(4 - 1) = 9

, so

(4, 11)

fits

Answer: $y - 2 = 3(x - 1)$

Example 4: convert to slope-intercept form

Rewrite $y + 1 = 2(x - 3)$ in slope-intercept form.

Start with the equation

y + 1 = 2(x - 3)

Distribute the slope

y + 1 = 2x - 6

Subtract

1

from both sides

y = 2x - 7

Check the point

(3, -1)

2(3) - 7 = -1

, so it's still on the line

Answer: $y = 2x - 7$

Try one yourself

Common questions

Which point should I use if I know two points on the line?

Either one. Both choices are correct equations for the same line — they just look different. In Example 3, using $(4, 11)$ instead gives $y - 11 = 3(x - 4)$ , and both simplify to the same slope-intercept form, $y = 3x - 1$ .

What point does

y + 6 = 5(x + 2)

pass through?

$(-2, -6)$ . Match the equation to the form $y - y_1 = m(x - x_1)$ : rewriting $y + 6$ as $y - (-6)$ and $x + 2$ as $x - (-2)$ shows $y_1 = -6$ and $x_1 = -2$ . Plus signs in the equation mean negative coordinates.

When should I use point-slope form instead of slope-intercept form?

Use point-slope form when you know a point that is not the $y$ -intercept — it takes the given information directly with no solving. Slope-intercept form is better for reading off the $y$ -intercept or comparing two lines quickly. Most problems end by converting from one to the other anyway.

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