Point slope form
Point slope form gives an equation of a line when one point on the line and the slope are known. The standard statement is \(y - y_1 = m(x - x_1)\), where the fixed point is \((x_1, y_1)\) and the slope is \(m\).
Core relationship. A line with slope m passing through (x1, y1) satisfies
\[\boxed{\,y - y_1 = m(x - x_1)\,}\]
Every \((x,y)\) on the line makes the “change in \(y\)” equal to \(m\) times the “change in \(x\)”.
Meaning from the slope definition
The slope between two distinct points \((x_1,y_1)\) and \((x,y)\) on the same non-vertical line is \[ m = \frac{y - y_1}{x - x_1}\quad (x \ne x_1). \] Multiplying by \((x - x_1)\) yields \(y - y_1 = m(x - x_1)\), which is exactly point slope form.
Equivalent line forms and conversions
Point slope form is one of several equivalent ways to write the same linear equation. Expanding or rearranging changes the appearance but not the geometric line.
| Form | Equation | What is immediately visible |
|---|---|---|
| Point slope form | \(y - y_1 = m(x - x_1)\) | A specific point \((x_1,y_1)\) and slope \(m\) |
| Slope-intercept form | \(y = mx + b\) | Slope \(m\) and \(y\)-intercept \(b\) |
| Standard form | \(Ax + By = C\) | Integers \(A,B,C\) often preferred for systems and elimination |
Expanding point slope form gives slope-intercept form: \[ y - y_1 = m(x - x_1) \;\Longrightarrow\; y = mx + (y_1 - mx_1). \] The intercept is \(b = y_1 - mx_1\). Rearranging further provides standard form \(Ax + By = C\) by clearing denominators and collecting terms.
Two points and point slope form
When two points \((x_1,y_1)\) and \((x_2,y_2)\) are known with \(x_2 \ne x_1\), the slope is \[ m = \frac{y_2 - y_1}{x_2 - x_1}. \] Substituting this \(m\) into point slope form produces a valid equation for the line through both points.
Special cases
Vertical lines have undefined slope and do not fit point slope form. A vertical line through \((x_1,y_1)\) is described by \[ x = x_1, \] which fixes the \(x\)-coordinate for every point on the line.
Worked example
A line passes through \((-2, 5)\) with slope \(m = \frac{3}{4}\). Point slope form is \[ y - 5 = \frac{3}{4}(x - (-2)) = \frac{3}{4}(x + 2). \] Expanding produces slope-intercept form: \[ y - 5 = \frac{3}{4}x + \frac{3}{2} \;\Longrightarrow\; y = \frac{3}{4}x + \frac{13}{2}. \] Clearing denominators gives a standard form: \[ 4y = 3x + 26 \;\Longrightarrow\; 3x - 4y = -26. \]
Common pitfalls
Sign errors in \(x - x_1\) and \(y - y_1\) are the most frequent source of incorrect equations; using \(x - (-2)\) as \(x + 2\) and \(y - 5\) exactly as written keeps the point anchored.
Fractional slopes require consistent distribution: \(m(x - x_1)\) means the slope multiplies the entire parenthesis.
Vertical lines correspond to \(x = x_1\) rather than point slope form because the slope is undefined.