# Point Slope Form Lesson

Point slope form is used to represent a linear equation, it is most useful when given a point on the line and the slope of the line. The point slope form equation is y – y_{1} = m (x – x_{1}), where (x_{1}, y_{1}) is the point on the line, m and is the slope of the line.

You can use point slope form when given a point on the line and its slope, you just plug in the values from the ordered pair for (x_{1}, y_{1}) and the slope m into the equation and simplify. This is just one method for finding the equation of a line.

For example, if we were given that the slope of a line is 4 and a point on the line is (2, 8), we could plug those values into the point-slope form and get the equation of the line, as seen below.

You can then simplify the above equation, and write the equation of the line in slope-intercept form as shown.

**A few reminders about equations of a line:**

- The equation of a horizontal line passing through a given point (m, n) will be of the form y = n.
- The equation of a vertical line passing through a given point (m, n) will be of the form x = m.
- Point slope form cannot be used to find the equation of a vertical line, since the slope is undefined.

**Steps for Using Point Slope form to find the equation of the line.**

1) Identify the slope (m), and the given point (x_{1} , y_{1}).

2) Substitute the values from Step 1 into the Point Slope Form y- y_{1} = m(x- x_{1}).

3) Distribute the slope into the parentheses.

4) Then solve the equation for y, to get the equation in slope-intercept form, y = mx + b.

Examples

**Example 1:** Find the equation of the line with a slope of m = -2 passing through the point (3, 5).

**Solution: **Using point-slope form: y – y_{1} = m (x – x_{1})

Substitute in the given values: y – 5 = -2(x – 3)

Distribute the -2 into the parentheses and solve for y.

y – 5 = -2x + 6

*Note that a negative times a negative is a positive, hence -2 x -3 = 6

Now we will add 5 to both sides to isolate the y, and write the equation in slope intercept form.

y = -2x + 11

Therefore, the equation in point-slope form is y – 5 = -2(x – 3) or written in slope intercept form y = -2x + 11

**Example 2: **Find the equation of a line with slope m = 1/3 passing through the point (9, -1).

**Solution:** Using point-slope form: y – y_{1} = m (x – x_{1})

Substitute in the given values: y – (-1) = 1/3 (x – 9)

*Note that when we subtract a negative, that becomes addition. Therefore, y – (-1) = y + 1

Distribute the 1/3 into the parentheses and solve for y.

y + 1 = 1/3x – 3

Now we will subtract 1 from both sides to solve for y, and get the equation:

y = 1/3x – 4

Therefore, the equation in point slope form is y + 1 = 1/3 (x – 9) and written in slope intercept form it is y = 1/3x -4

**Example 3: **Find the equation of the line given the slope is – 5/2 and goes through the point (-6, 5).

**Solution: **Start by identifying the slope and given point, and substitute them into point slope form. We know that our slope is – 5/2 which is our m and the given point is our (x_{1}, y_{1}). Therefore, we have

y – 5 = -5/2 (x – (-6))

*Note that when we subtract a negative, that becomes addition.

y – 5 = -5/2 (x + 6)

Then we will distribute the -5/2, to get

y – 5 = -5/2x – 15

Now we will add the 5 to both sides to solve for y, and get the equation:

y = -5/2x – 10

Therefore, the equation in point slope form is y – 5 = -5/2 (x + 6), and written in slope intercept form is

y = -5/2x – 10

FAQs on Point Slope Form

**What is the point-slope form?**

Point-slope form is a way to represent the equation of a straight line. It uses a point (x_{1}, y_{1}) on the line and the slope m of the line to express the equation as: y- y_{1} = m(x- x_{1}).

**How is point-slope form different from other forms of linear equations?**

Point-slope form is similar to slope-intercept form (y = mx+b), but instead of giving the y-intercept directly, it uses a specific point on the line. This makes it convenient when you have a point and the slope of the line, but don’t necessarily know the y-intercept.

**When should I use point-slope form?**

Point-slope form is useful when you have a specific point on the line and the slope of the line. If you’re given these pieces of information and need to find the equation of the line, point-slope form is often the most straightforward method to use.

**How do I convert an equation from point-slope form to slope-intercept form?**

To convert an equation from point-slope form to slope-intercept form, simply distribute the slope and then solve for y. The equation will end up in the form y = mx+b, where m is the slope and b is the y-intercept.

**Can I find the equation of a line if I only have one point?**

No, to uniquely determine the equation of a line, you need both the slope of the line and one point that lies on it. With just one point, there are infinitely many lines that could pass through it, so you need the slope to narrow down the possibilities.

**Is point-slope form the only way to represent a linear equation?**

No, there are other forms such as slope-intercept form, standard form, and general form. Each form has its own advantages and is useful in different situations. Point-slope form is particularly useful when you have a specific point and the slope of the line.

**How do you find the Point Slope Form from a Graph?**

To find point slope form from a graph, identify a point on the line and then determine the slope of the line either by using two points on the line, or from determining the rise over the run. Once you have the slope and a point on the line, you can write the equation in Point Slope form.