Point Slope Intercept Calculator: Find Line Equation

This calculator finds the equation of a line in point-slope and slope-intercept forms. You enter the slope and one point on the line (x�, y�), and it returns the line’s equation in both y – y� = m(x – x�) and y = mx + b forms, along with the y-intercept and x-intercept.

Formula

The point-slope form of a linear equation is:

y – y� = m(x – x�)

where:

• m = slope of the line (unitless)
• (x�, y�) = any known point on the line (in the same coordinate units as x and y)
• x, y = variables representing any other point on the line

To convert to slope-intercept form (y = mx + b), solve for y:

y = mx – m·x� + y�

So the y-intercept is b = y� – m·x�.

Worked Example

Using the default values from the calculator above: m = 2, point (3, 5).

Point-slope form:

y – 5 = 2(x – 3)

Now solve for y:

y – 5 = 2x – 6

y = 2x – 1

So the y-intercept b = –1, and the x-intercept (where y = 0) is:

0 = 2x – 1 → x = 0.5

The calculator shows the same: point-slope form “y – 5 = 2(x – 3)�, slope-intercept form “y = 2x + -1�, y-intercept –1.00, x-intercept 0.50.

When to Use Point-Slope vs Slope-Intercept

Point-slope form is most useful when you already know the slope and one point on the line – for example, from a word problem or a graph. It is also the quickest way to write an equation when you have two points but not the y-intercept. Slope-intercept form (y = mx + b) is better for quickly reading the slope and y-intercept, graphing, and comparing lines. Many textbooks prefer slope-intercept for final answers, so you may need to convert.

How to Use the Calculator

Simply enter the slope m and the coordinates of one point (x�, y�). The calculator instantly gives you:

• Point-slope equation
• Slope-intercept equation
• y-intercept (where the line crosses the y-axis)
• x-intercept (where the line crosses the x-axis)

You can use any real numbers, including fractions (as decimals) and negative values. The calculator handles them all.

Common Mistakes and Misconceptions

A frequent error is mixing up x� and y� when substituting. Always attach the y-coordinate to the y side of the equation. Another common mistake is forgetting to distribute the slope when converting to slope-intercept form – for example, writing y – 5 = 2x – 3 instead of y – 5 = 2x – 6. Also, remember that the slope m is the same regardless of which point you choose, but the point-slope form will look different if you pick a different point (though it represents the same line).

Limitations of the Formula

The point-slope formula only works for non-vertical lines. A vertical line has an undefined slope and cannot be expressed in point-slope or slope-intercept form; its equation is x = constant. Also, the formula assumes a perfectly linear relationship – it cannot describe curves. For real-world data, the slope-intercept form is often used in linear regression, but the point-slope form is rarely used for statistical fitting because it requires a known point.

Related Forms and Conversions

Besides point-slope and slope-intercept, you may encounter standard form: Ax + By = C. To convert from slope-intercept to standard form, rearrange and clear fractions. For example, y = 2x – 1 becomes 2x – y = 1. The point-slope form is particularly handy when you have two points: first compute the slope using m = (y₂ – y�) / (x₂ – x�), then plug one point into the point-slope formula.

Frequently Asked Questions

What is the point-slope formula?

The point-slope formula is y – y� = m(x – x�), where m is the slope and (x�, y�) is a known point on the line. It is used to write the equation of a line when you know the slope and one point.

How do I convert point-slope to slope-intercept?

Solve the point-slope equation for y. Distribute the slope on the right side, then add y� to both sides. The result will be y = mx + (y� – m·x�), where the term in parentheses is the y-intercept b.

Can I use the point-slope form with two points?

Yes. First calculate the slope using m = (y₂ – y�) / (x₂ – x�), then choose either point and plug it into the point-slope formula. Both points will yield the same line.

What if the slope is zero or undefined?

If the slope is zero, the line is horizontal and the equation is y = y�. If the slope is undefined (vertical line), the point-slope form cannot be used; the equation is x = x�.

Why do I get a different point-slope equation when I use the other point?

Because the point-slope form is written relative to a specific point. Using a different point changes the left side (y – y�) and the right side (x – x�), but the equation still represents the same line. The slope-intercept form will be identical either way.

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in m | Slope (m) | - | 2 | 0
in x1 | Point x� | - | 3 | 0
in y1 | Point y� | - | 5 | 0
out eq_point | Point-slope form | - | "y - " + y1 + " = " + m + "(x - " + x1 + ")" | 0
out eq_slope | Slope-intercept form | - | "y = " + m + "x + " + (y1 - m*x1) | 0
out b | y-intercept (b) | - | y1 - m*x1 | 2
out x_int | x-intercept | - | -b / m | 2