How do you convert point slope form to slope-intercept form?

Start with the equation that states that the difference between y and y1 is equal to the product of the slope, m, and the difference between x and x1. Distribute the slope in parentheses so that the difference between y and y1 is equal to the difference between mx and mx1. Then add y1 to both sides of the equation. The result is that y equals mx plus b where b is equal to the difference between y1 minus m times x1. The slope remains the same, and the constant b is the y-intercept.

How do you find the equation of a line given two points?

First calculate the slope by dividing the change in y by the change in x. That is to say, divide the difference between (y2 and y1) by the difference between (x2 and x1). Then plug that slope into the point-slope formula along with one of the points you were given. For example, if a line has a slope of negative three halves and passes through the points (1, 2) and (5, -4), then its equation is y minus 2 equals negative three halves times x minus 1.

What does the point-slope formula look like when the slope is zero?

When the slope is zero, the line is horizontal. The point-slope formula becomes y minus y1 equals zero, which means that y is equal to y1, a constant. The line intersects the y-axis at y1 and never crosses the x-axis unless it was originally on the x-axis. This calculator shows a slope of zero as a horizontal line.

Why is there no point-slope form for vertical lines?

Since all points on a vertical line have the same x-value, the change in the x-coordinate between any two points is zero. The slope is calculated by dividing the change in the y-coordinate by the change in the x-coordinate, but division by zero is undefined. Therefore, there is no slope for vertical lines, and thus no point-slope form or slope-intercept form of a line. The equation of such a line takes the form x equals some constant. If the x-values are equal when using the two-point method, then the calculator will recognize that the line is vertical.

What are the relationships between point-slope form, slope-intercept form, and standard form?

These are three different ways to represent the same line. The point-slope form is made up of a point and the slope. The slope-intercept form directly shows the slope and y-intercept. The standard form uses integer coefficients. Any one form can be converted into any other, and this calculator will show all three forms at once.

Point-Slope Form Calculator

Enter the line

What do you know?

Pick how you want to define the line.

A point and the slope

Two points on the line

The point the line passes through, written (x₁, y₁).

Slope m

Find another point on the line

Enter any x and read off the matching y on the line.

The line's equation

Point-slope form
y + 1 = 3(x - 2)
Slope-intercept form
y = 3x - 7
Standard form
3x - y = 7

y-intercept b

Point-slope form
y + 1 = 3(x - 2)

Slope (m)
y-intercept (b)
x-intercept

Angle of inclination
Perpendicular slope

The slope you entered is m = 3.

A parallel line has the same slope, 3. A perpendicular line has the negative reciprocal slope, -0.333333.

Graph and steps

Show the line graph

Plot the line and see where it crosses the axes.

Show the step-by-step working

Walk through slope, point-slope, and the rearranged forms.

From your inputs to the line in all three forms
Step	Working
Identify the point and the slope	point (2, -1), m = 3
Write point-slope form y - y1 = m(x - x1)	y + 1 = 3(x - 2)
Solve for y -> b = y1 - m*x1	b = -1 - (3)(2) = -7
Slope-intercept form y = m x + b	y = 3x - 7
Rearrange to standard form Ax + By = C	3x - y = 7

The point-slope form is the fastest method to determine the equation of a line when one point on the line and its slope are known. This calculator accepts such a point and such a slope as input. If two points are known, they can be entered directly. The tool will then output not only the point-slope form but also the slope-intercept form and the general form of the line. In addition, it calculates the slope, both intercepts, and the angle between the line and the x-axis.

This tool is also suitable for cases where errors can easily be made. The slope of a vertical line is undefined, so there is no point-slope form for it. The slope of a horizontal line is zero. This calculator recognizes these two cases and displays the correct equation in each case.

What is point-slope form?

The point-slope form is a way to express the equation of a line in terms of three variables: the coordinates of a point on the line and its slope. The formula for the point-slope form is: y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope.

y - y_1 = m\,(x - x_1)

This is directly from the definition of slope. The slope is the change in y over the change in x between any two points. So it's the change in y divided by the change in x between any two points.

m = \frac{\text{change in } y}{\text{change in } x} = \frac{y - y_1}{x - x_1}

When both sides are multiplied by the change in x, it becomes point-slope form. This is why the two formulas look so similar. Point-slope form is just a rearranged version of the slope formula with the denominator eliminated.

This is how this calculation tool works:

Select the known information. If you have a point and a slope, enter the point as x1 and y1 and then enter the slope. If you have two points, switch to the two-point mode and enter both points. The calculator will first calculate the slope and then generate all of the formulas based on that.

For coordinates and slopes you can enter decimals. A slope of three halves can be entered as 1.5 and a negative slope of three quarters as -0.75. If possible, the result is shown as a reduced fraction instead of a long decimal (e.g. three halves instead of a long decimal).

By opening the additional point finder tool, you can read off the y-value for any x-value on the line. By opening the graph, you can see the line in relation to the coordinate axes. By expanding the steps, you can follow the calculation process from the slope to the general form.

Transformation to three different forms.

There are three forms that a linear equation can take; each is useful under different circumstances. The point-slope form is the fastest way to write an equation for a line. In slope-intercept form, you can immediately read off the slope and y-intercept of the line. Standard form uses only integers, which makes it great for many applications in textbooks.

To convert from point-slope form to slope-intercept form, you want to get the equation into the form y = mx + b by solving for y. The y-intercept can be found directly from the point and the slope.

b = y_1 - m\,x_1 \quad\Rightarrow\quad y = m\,x + b

To get a general form, combine x and y so that their coefficients are integers, and bring the constants to the other side.

A\,x + B\,y = C

By convention, A should not be negative and the three coefficients should have no common divisor other than one; thus, 3x + 2y = 7 is written rather than 6x + 4y = 14.

A complete example:

Take the point (2, -1) and a slope of 3. If you plug those into the point-slope formula and remember that subtracting a negative is the same as adding, you get this:

y - (-1) = 3\,(x - 2) \quad\Rightarrow\quad y + 1 = 3\,(x - 2)

Solving for y will give you slope-intercept form and moving everything to one side will give you standard form.

y = 3x - 7 \qquad 3x - y = 7

In the case of two points, the additional step is simply to calculate the slope. For the points (1, 2) and (5, -4), the slope is the change in y-coordinates divided by the change in x-coordinates.

m = \frac{-4 - 2}{5 - 1} = \frac{-6}{4} = -\frac{3}{2}

With the first point this line can be represented as an equation where the difference of y and 2 is equal to the product of minus three halves and the difference of x and 1. Simplified, it results in: y is equal to minus three halves x plus seven halves, or in general form: 3x + 2y = 7.

Vertical and Horizontal Lines

Two special cases often cause confusion. A vertical line goes straight up and down, so all points on it have the same x-value and there is no change in the x-coordinate between any two points. The slope is the ratio of the change in y-coordinates to the change in x-coordinates, but division by zero is undefined. So a vertical line has no slope and cannot be written in point-slope form. Its equation is of the form x = a constant, such as x = 4.

A horizontal line is the opposite. Since it does not move up at all, there is no change in y and therefore the slope is zero. The point-slope form can still be used and simplifies to the form y = a constant, such as y = 3. This calculator recognizes when x2 is equal to x1 as a vertical line, and when the slope is zero as a horizontal line, and will label it accordingly.

Parallel and perpendicular lines

The slope is also important for determining related lines. If two lines have the same slope they are parallel. So a line that is parallel to a line with a slope of m will also have a slope of m. If two lines are perpendicular their slopes are negative reciprocals of each other. This means that the slope of one line is equal to the negative reciprocal of the slope of the other line.

A line with a slope of 2 is perpendicular to a line with a slope of -1/2 because the product of the two slopes is -1. A calculator can show you the slope of a perpendicular line, so that you can create a perpendicular line through any given point.

Form	Equation	Best for
Point-slope	$y - y_1 = m(x - x_1)$	Writing a line from a point and slope
Slope-intercept	$y = mx + b$	Reading off the slope and y-intercept
Standard	$Ax + By = C$	Integer coefficients, comparing lines

Frequently asked questions

What is point-slope form?: The point-slope form is a way to write the equation of a line that involves a known point on the line and the slope of the line. The difference between y and y1 is equal to the product of the slope m and the difference between x and x1. Where (x1, y1) are the coordinates of the known point and m is the slope. This comes directly from the formula for slope, which is change in y divided by change in x. Since the denominator cancels out this is the fastest way to write an equation when you know a point and the slope.
How do you convert point slope form to slope-intercept form?: Start with the equation that states that the difference between y and y1 is equal to the product of the slope, m, and the difference between x and x1. Distribute the slope in parentheses so that the difference between y and y1 is equal to the difference between mx and mx1. Then add y1 to both sides of the equation. The result is that y equals mx plus b where b is equal to the difference between y1 minus m times x1. The slope remains the same, and the constant b is the y-intercept.
How do you find the equation of a line given two points?: First calculate the slope by dividing the change in y by the change in x. That is to say, divide the difference between (y2 and y1) by the difference between (x2 and x1). Then plug that slope into the point-slope formula along with one of the points you were given. For example, if a line has a slope of negative three halves and passes through the points (1, 2) and (5, -4), then its equation is y minus 2 equals negative three halves times x minus 1.
What does the point-slope formula look like when the slope is zero?: When the slope is zero, the line is horizontal. The point-slope formula becomes y minus y1 equals zero, which means that y is equal to y1, a constant. The line intersects the y-axis at y1 and never crosses the x-axis unless it was originally on the x-axis. This calculator shows a slope of zero as a horizontal line.
Why is there no point-slope form for vertical lines?: Since all points on a vertical line have the same x-value, the change in the x-coordinate between any two points is zero. The slope is calculated by dividing the change in the y-coordinate by the change in the x-coordinate, but division by zero is undefined. Therefore, there is no slope for vertical lines, and thus no point-slope form or slope-intercept form of a line. The equation of such a line takes the form x equals some constant. If the x-values are equal when using the two-point method, then the calculator will recognize that the line is vertical.
What are the relationships between point-slope form, slope-intercept form, and standard form?: These are three different ways to represent the same line. The point-slope form is made up of a point and the slope. The slope-intercept form directly shows the slope and y-intercept. The standard form uses integer coefficients. Any one form can be converted into any other, and this calculator will show all three forms at once.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Khan Academy: Point-slope form
Lessons on point-slope, slope-intercept, and standard forms of linear equations.
Wolfram MathWorld: Line
Formal treatment of the equation of a line, including two-point and point-slope forms.