Point-Slope Form Calculator

About Point-Slope Form Calculator

The Point-Slope Form Calculator finds the equation of a line given a point and a slope, or given two points. It outputs the equation in three standard formats — point-slope form, slope-intercept form, and standard form — with a step-by-step solution and interactive coordinate plane graph.

How to Use the Point-Slope Form Calculator

1. Choose your input mode: Select "Point & Slope" if you know one point and the slope, or "Two Points" if you have two points on the line.
2. Enter the coordinates: Type the \(x\) and \(y\) values for your known point(s). Use the parenthesized input fields for intuitive coordinate entry.
3. Enter the slope (if applicable): Type the slope as a decimal (e.g., 0.5) or a fraction (e.g., 2/3). Negative slopes work too (e.g., -3/4).
4. Click "Calculate Equation" to see the results instantly.
5. Review the output: Three equation cards show the line in point-slope, slope-intercept, and standard form. Use the copy buttons to grab any equation. Scroll down for the step-by-step solution, line properties, and interactive graph.

What Is Point-Slope Form?

Point-slope form is a way of writing the equation of a straight line. If you know a point \((x_1, y_1)\) on the line and the slope \(m\), the equation is:

$$y - y_1 = m(x - x_1)$$

This form is especially useful when you don't know the y-intercept directly. It's derived from the definition of slope: \(m = \frac{y - y_1}{x - x_1}\).

Converting Between Forms

Point-Slope to Slope-Intercept Form

Starting from \(y - y_1 = m(x - x_1)\):

1. Distribute: \(y - y_1 = mx - mx_1\)
2. Add \(y_1\): \(y = mx - mx_1 + y_1\)
3. The result is \(y = mx + b\) where \(b = y_1 - mx_1\)

Slope-Intercept to Standard Form

From \(y = mx + b\):

1. Rearrange: \(-mx + y = b\), or equivalently \(mx - y = -b\)
2. If \(m\) is a fraction, multiply through to clear denominators
3. The result is \(Ax + By = C\) with \(A \geq 0\)

Using Two Points

If you have two points \((x_1, y_1)\) and \((x_2, y_2)\), first compute the slope:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Then substitute the slope and either point into the point-slope formula. Both points give the same line.

Understanding the Graph

The interactive graph shows:

• The line drawn with an animation across the coordinate plane
• Your input point(s) marked with colored dots and labeled coordinates
• The slope triangle (rise over run) near your point, showing the geometric meaning of slope
• Intercepts: the y-intercept (green dot) and x-intercept (orange dot) where applicable

Special Cases

• Horizontal line (m = 0): The equation simplifies to \(y = y_1\), a constant.
• Slope of 1: The line makes a 45-degree angle with the x-axis.
• Negative slope: The line falls from left to right.
• Fractional slope: Enter as a/b (e.g., 2/3). The calculator handles fractions natively.
• Vertical lines have undefined slope and cannot be expressed in point-slope form. If your two points share the same x-coordinate, the calculator will alert you.

FAQ

What is point-slope form?

Point-slope form is a way to write the equation of a line when you know one point on the line and the slope. The formula is y - y1 = m(x - x1), where (x1, y1) is the known point and m is the slope.

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

Distribute the slope m across (x - x1) to get y - y1 = mx - mx1. Then add y1 to both sides: y = mx - mx1 + y1. The constant term -mx1 + y1 is the y-intercept b, giving y = mx + b.

Can you use two points instead of a point and slope?

Yes. First calculate the slope using m = (y2 - y1) / (x2 - x1), then plug the slope and either point into the point-slope formula y - y1 = m(x - x1).

What is standard form of a linear equation?

Standard form is Ax + By = C, where A, B, and C are integers and A is non-negative. It is useful for finding intercepts and for systems of equations.

What if the slope is a fraction?

You can enter fractions directly as a/b, for example 2/3 or -3/4. The calculator handles fractions and displays them properly in the results.

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