Convert a point slope equation
Use decimals, integers, or fractions such as -3/4. Required fields are marked.
Formula Used
The starting equation is y − y₁ = m(x − x₁). Here, m is the slope. The ordered pair is (x₁, y₁).
Expand the right side first. Then add y₁ to both sides. The result becomes y = mx + (y₁ − mx₁).
The y-intercept is b = y₁ − mx₁. It is the point where the line crosses the y-axis. At that location, x equals zero.
How to Use This Calculator
- Enter the x-coordinate from the known point.
- Enter the y-coordinate from the same point.
- Enter the line slope, including its sign.
- Optionally add another x-value to find its y-value.
- Select Convert Equation to view forms and checks.
- Use the download buttons to save the completed result.
Worked Example Data
| Input | Value | Calculation | Output |
|---|---|---|---|
| Point | (2, 5) | y − 5 = 3(x − 2) | Point slope form |
| Slope | 3 | b = 5 − 3(2) | b = −1 |
| Y intercept form | — | y = 3x − 1 | (0, −1) |
| Verification | x = 2 | 3(2) − 1 | y = 5 |
Move From Point Slope Form to Y Intercept Form
Point slope form describes a line from one known point and its slope. It is compact. It also keeps the original information visible. Y intercept form is often easier to graph. It shows the slope and the vertical crossing immediately. Both forms describe exactly the same line.
Start with y minus y₁ equals m times x minus x₁. The symbols x₁ and y₁ represent a point on the line. The letter m represents slope. Slope measures vertical change for each horizontal change. A positive slope rises from left to right. A negative slope falls from left to right.
Expand Carefully
Multiply the slope by every term inside the parentheses. A common mistake is forgetting the negative sign before x₁. For example, 4(x − 3) becomes 4x − 12. When the point uses a negative coordinate, keep its parentheses. For example, x minus negative two becomes x plus two.
After expanding, add y₁ to both sides. Place the x term first. Combine constant values last. The completed structure is y equals mx plus b. The constant b is the y-intercept. It tells you where the line meets the vertical axis.
Find and Check the Intercept
The calculator uses b equals y₁ minus m times x₁. This shortcut gives the intercept directly. It matches the algebraic expansion. It is useful when you need a fast check. Enter the original point and slope. The result shows b, y intercept form, standard form, and a substitution check.
Always test the converted equation with the original point. Replace x with x₁. The calculated y should equal y₁. This catches sign errors early. You can also set x to zero. The resulting y should equal b. That second check confirms the intercept coordinate.
Use Fractions and Decimals Correctly
Lines may use whole numbers, decimals, or fractions. Fractions are accepted in this calculator. Enter 3/4 rather than changing it to a rounded decimal. Exact values often make schoolwork cleaner. Negative fractions work too. Enter them with the negative sign before the numerator.
Decimals are useful for measured data. Keep enough digits when accuracy matters. The result is displayed neatly, without unnecessary trailing zeros. Very large values can make manual checking harder. Work slowly and inspect every sign. A small sign error changes the line completely.
Graph the Final Equation
Once you have y equals mx plus b, plot the intercept first. Then use the slope as rise over run. For a slope of two thirds, go up two and right three. For a negative slope, move down or left as needed. Draw a straight line through the points.
This tool supports homework checks, tutoring practice, and quick conversions. It does not replace learning the steps. Practice with several examples until each algebraic move feels natural and predictable. Read the formula, inspect the expansion, and verify the result. Those habits make linear equations easier to solve, graph, and explain.
Frequently Asked Questions
1. What is point slope form?
Point slope form is y − y₁ = m(x − x₁). It defines a line with one known point and its slope. The symbols x₁ and y₁ identify the point. The letter m identifies the slope.
2. What is y intercept form?
Y intercept form is y = mx + b. The slope is m. The constant b is the y-intercept. This form helps you graph because the starting point on the y-axis is visible.
3. How do I find b from a point and slope?
Use b = y₁ − mx₁. Multiply the slope by the point’s x-coordinate. Then subtract that product from the point’s y-coordinate. The answer is the y-intercept.
4. Can the slope be zero?
Yes. A zero slope makes a horizontal line. The equation becomes y = b. Every point on that line has the same y-coordinate, while x may change.
5. Can I enter a negative point?
Yes. Negative coordinates are valid. Keep the sign with the coordinate. Parentheses in point slope form help prevent errors when a coordinate is negative.
6. Are fractions accepted?
Yes. Enter fractions such as 2/3, -5/4, or 1/2. Do not use a zero denominator. The calculator converts accepted fractions to numeric values for the result.
7. Why is my intercept negative?
A negative intercept simply means the line crosses below zero on the y-axis. It is normal. Check the calculation b = y₁ − mx₁ to confirm the sign.
8. How can I verify the final equation?
Substitute the original x-coordinate into y = mx + b. The output should equal the original y-coordinate. You can also set x to zero and confirm that y equals b.
9. What does the optional x-value do?
It finds another point on the same line. The calculator substitutes your optional x-value into the completed y intercept equation and returns its matching y-value.
10. Does standard form represent the same line?
Yes. Standard form rearranges the same equation, commonly as Ax + By + C = 0. The displayed standard form is equivalent to the y intercept result.
11. Can this calculator replace showing work?
It is best used to check work and understand each step. Many teachers require the expansion and rearrangement process. Use the displayed formulas to explain your reasoning clearly.