How to Solve Point-Slope Form of Equations?

The equation of a line can be found in different ways depending on the available information. Some methods include point-slope form, slope-intercept form, intercept form, and two-point form.

Related Topics

• How to Find Slope
• How to Graph Lines by Using Slope–Intercept Form

A step-by-step guide to point-slope form

The point-slope form is used to represent a straight line using its slope and a point on the line. That is, the equation of a line whose slope is \(m\) and passes through a point \((x_1,y_1)\) is found using the point-slope form.

Different shapes can be used to express the equation of a straight line. One of them is the point-slope form. The equation of the point-slope form is:

\(\color{blue}{y-y_1=m(x-x_1)}\)

where,

• \((x, y)\) is a random point on the line (which should be kept as variables while applying the formula).
• \((x_1, y_1)\) is a fixed point on the line.
• \(m\) is the slope of the line.

Note: This formula is used only when we know the slope of the line and a point on the line.

How to solve point-slope form?

To solve the point-slope form for a given straight line to find the equation of the given line, we can follow these steps:

• Step 1: Note down the slope, \(m\) of the straight line, and the coordinates \((x_1,y_1)\) of the given point that lies on the line.
• Step 2: Substitute the given values in the point-slope formula.
• Step 3: Simplify to get the equation of the line in standard form.

Point-Slope Form – Example 1:

Find the equation of a line that passes through a point \((2, -4)\) and whose slope is \((-\frac{1}{2})\).

Solution:

The point on the given line is: \((x_1, y_1) = (2, -4)\)

The slope of the line is: \(m = (-\frac{1}{2})\)

The equation of the line is found using the point-slope form:
\(y − y_1= m(x − x_1)\)
\(y− (−4) = (−\frac{1}{2})(x − 2)\)
\(y+4= (−\frac{1}{2})x+ 1\)

Subtracting \(4\) from both sides:
\(y = (−\frac{1}{2})x− 3\)

Exercises for Point-Slope Form

1. Find the equation of a horizontal line that passes through a point \((4, 3)\).
2. Find the equation of a line that passes through two points \((1, 2)\) and \((-3, 4)\) using the point-slope form.

1. \(\color{blue}{y=3}\)
2. \(\color{blue}{x=-2y+5}\)