# 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.

## 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. $$\color{blue}{x=-2y+5}$$

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