Equation of a Line (Point–Slope)

Statement

The equation of a straight line passing through a point \((x_1,y_1)\) with slope (gradient) \(m\) is:

\[ y - y_1 = m(x - x_1) \]

This is known as the point–slope form of a line.

Why it’s true

• Slope \(m\) is defined as the change in \(y\) over the change in \(x\): \(\frac{y-y_1}{x-x_1} = m\).
• Rearranging gives \(y-y_1=m(x-x_1)\).
• So this equation captures all points \((x,y)\) that make the slope equal to \(m\) when connected to \((x_1,y_1)\).

Recipe (how to use it)

1. Identify the slope \(m\) and a point \((x_1,y_1)\) on the line.
2. Substitute into \(y-y_1=m(x-x_1)\).
3. Simplify into slope–intercept form \(y=mx+c\) if needed.

Spotting it

This form is useful when you know a point on the line and the slope, but not the intercept.

Common pairings

• Slope–intercept form \(y=mx+c\).
• General linear form \(ax+by+c=0\).
• Parallel and perpendicular line problems.

Mini examples

1. Given: Point (2,3), slope 4. Equation: \(y-3=4(x-2)\) → \(y=4x-5\).
2. Given: Point (-1,2), slope -½. Equation: \(y-2=-\tfrac{1}{2}(x+1)\) → \(y=-\tfrac{1}{2}x+1.5\).
3. Given: Point (0,5), slope 3. Equation: \(y-5=3(x-0)\) → \(y=3x+5\).

Pitfalls

• Mixing up \((x,y)\) (variable coordinates) with \((x_1,y_1)\) (fixed point).
• For vertical lines slope is undefined, so this form does not apply.

Exam strategy

• Always substitute the given point into \((x_1,y_1)\) carefully.
• Convert into slope–intercept form if asked for the equation in \(y=mx+c\).
• For perpendicular lines, remember slope is negative reciprocal.

Summary

The point–slope form of a line \(y-y_1=m(x-x_1)\) is powerful when you know one point and the slope. It can easily be transformed into other linear equation forms.