# How to Solve Point-Slope Form of Equations?

The equation of a line with a definite slope and with a given point is found using the point-slope form. In this step-by-step guide, you learn more about finding the equation of a line using the point-slope form.

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**

## 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 **

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

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

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