How to solve point slope form? To find the equation of a straight line that passes through a given point and is inclined at a given angle to the x-axis, use the point-slope form. Each and every point on a line is able to satisfy the equation of the line. So, a two-variable linear equation represents the line. The equation of a line can be determined in a variety of ways depending on the information available. These methods include:

• Point slope form
• Slope-intercept form
• Intercept form
• Two-point form

We use the point-slope formula only when we know the slope of the line and a point on it. In the following section, we will examine the point-slope form in more detail and learn how to derive the formula to represent the point-slope form.

Point Slope Form: What is it?

A straight line’s slope and a point on the line are used to represent its slope in point-slope form. Thus, the equation of a line whose slope is ‘m’ and passes through the point (x11, y11) is found using the point-slope form. There are several ways to express the equation of a straight line. A point-slope form is one of them. A point-slope form has the equation:

y – y11 = m(x – x11)

(x, y) represents a random point on the line and m represents the slope.

Point Slope Formula

Using the point-slope form formula, you can find a line’s equation. When a slope and a point are given, the equation for the line can be found by using the point-slope form. When the slope of the line and a point on the line are known, we can apply this formula. As well as the slope-intercept form and the intercept form, there are some other formulas to calculate the equation of a line. These are the point-slope formula, the intercept formula, and the slope formula.

Point Slope Formula in Math:

y − y11 = m (x − x11)

where,

• (x, y) are the random points on the line (which should be kept as variables when applying the formula).
• (x11, y11) represents a fixed point along a line.
• m is the slope of the line.

Derivation of Point-Slope Formula

The point-slope form, or the proof of the formula for the point-slope form, can be found by following the steps below. Based on the equation for a line’s slope, we will derive this formula. For example, consider a line whose slope is m. We will assume that (x11, y11) is a known point along the line. Assume that (x, y) is another random point on the line whose coordinates are unknown.

A line’s slope can be calculated by using the following equation:

The slope is the ratio of (difference in y-coordinates) to (difference in x-coordinates).

m = (y – y11)/(x – x11)

Multiplying both sides by (x – x11),

m(x – x11) = y – y11

This can be written as,

y – y11 = m(x – x11)

Hence the point-slope formula is proved.

Point Slope Formula Examples

The point-slope formula is illustrated here with examples.

1. If there is a slope of (-1) and a point (1, 2), we can find the equation by using y – 2 = (-1)(x – 1).
2. To find the equation of a line with a slope of (3/2) and a point of (-1/2, 2/3), use the formula: y – (2/3) = (3/2) (x – (-1/2)).
3. In order to find an equation for a line with slope (0) and a point (3, -2) use the formula: y – (-2) = 0(x – 3).

With each of these cases, we can simplify the equation further and get it to look like this: y = mx + b.

Important notes on Point-Slope Form:

• Y – y11 = m(x – x11) is the equation of the point-slope form of a line with slope ‘m’ and that passes through the point (x11, y11).
• An equation for a horizontal line passing through (a, b) takes the form y = b.
• An equation of form x = a can be expressed as the equation of a vertical line passing through (a, b).

In this instance, it is not possible to use the point-slope form.

How to Solve Point-Slope Form?

For finding the equation of a given straight line, we can solve point-slope form using the steps below;

• Step 1: note the slope, ‘m’ of the straight line, and the coordinates (x11, y11) of the given point that lies on it.
• Step 2: Use the following formula to calculate the point slope: y – y11 = m(x – x11).
• Step 3: Simplify the line equation to obtain the standard form.