The point slope form is a standard algebraic form for a linear equation that uses one known point on the line and the slope. It is especially efficient when the slope \(m\) and a point \((x_1,y_1)\) are given directly.
Point slope form (definition):
\[ y - y_1 = m(x - x_1). \]
Here, \(m\) is the slope and \((x_1,y_1)\) is any point on the line.
Why the point slope form works
Slope is defined as “rise over run” between two points \((x_1,y_1)\) and \((x,y)\) on the same line:
\[ m=\frac{y-y_1}{x-x_1}\quad (x\ne x_1). \]
Multiply both sides by \((x-x_1)\) to isolate the vertical change:
\[ y-y_1=m(x-x_1), \]
which is exactly the point slope form. This equation represents all points \((x,y)\) that maintain the same slope relative to \((x_1,y_1)\).
How to write an equation using point slope form
Given a slope \(m\) and a point \((x_1,y_1)\), substitute into \(\,y-y_1=m(x-x_1)\,\) and simplify if desired.
Worked example
Construct the line with slope \(m=2\) passing through \((x_1,y_1)=(1,3)\). This is a typical point slope form situation.
\[ y-3 = 2(x-1). \]
Converting point slope form to slope-intercept form
Slope-intercept form is \(y=mx+b\). To convert, expand and solve for \(y\).
\[ y-3 = 2(x-1) \]
\[ y-3 = 2x-2 \]
\[ y = 2x+1 \]
General conversion rule: Starting from \(y-y_1=m(x-x_1)\), expand to get
\[ y = mx + (y_1 - mx_1). \]
Therefore the intercept is \(\,b = y_1 - m x_1\).
Quick comparison of common linear forms
| Form | Equation | Best used when |
|---|---|---|
| Point slope form | \(y-y_1=m(x-x_1)\) | A point and slope are given |
| Slope-intercept form | \(y=mx+b\) | Slope and y-intercept are needed quickly for graphing |
| Standard form | \(Ax+By=C\) | Integer coefficients are preferred or constraints are given in standard form |
Visualization: line from point slope form
The graph below shows the example line \(y-3=2(x-1)\), which simplifies to \(y=2x+1\). The marked point \((1,3)\) lies on the line, and a second point \((2,5)\) illustrates the slope \(m=2\) (rise \(2\), run \(1\)).
Common mistakes to avoid
- Mixing up the point: the form must be \(y-y_1=m(x-x_1)\). The subtraction signs are tied to the chosen point \((x_1,y_1)\).
- Sign errors when \(x_1\) or \(y_1\) is negative: for example, if \(x_1=-4\), then \(x-x_1=x-(-4)=x+4\).
- Confusing \(m\) with an x-intercept or y-intercept: in point slope form, \(m\) is strictly the slope.
Summary
The point slope form \(y-y_1=m(x-x_1)\) is derived directly from the slope definition and is ideal for writing a linear equation from a given point and slope. Expanding and isolating \(y\) converts it to slope-intercept form \(y=mx+b\) with \(b=y_1-mx_1\), which is often convenient for graphing and interpretation.