Coordinate Grid Calculator
Plot points on a coordinate grid, calculate distances, midpoints, slopes, areas, and identify quadrants. Includes SVG grid display, pairwise distance table, and presets.
Point-Slope Form Calculator
Convert between point-slope, slope-intercept, and standard form. Find the equation of a line from two points, parallel lines, and perpendicular lines with visual plots and comparison tables.
Slope (m)⧉
2.000000
Rise over run — how steep the line is
Y-Intercept (b)⧉
0.000000
Where the line crosses the y-axis (x = 0)
X-Intercept⧉
-0.000000
Where the line crosses the x-axis (y = 0)
Angle with X-axis⧉
63.43°
Acute angle between the line and the positive x-axis
Parallel Slope⧉
2.000000
Slope of any line parallel to this one (same m)
Perpendicular Slope⧉
-0.500000
Slope of a line perpendicular to this one (−1/m)
Distance Between Points⧉
6.708204
Euclidean distance √[(x₂−x₁)² + (y₂−y₁)²]
Line Equations — All Three Forms
| Form | Equation | When to Use |
|---|---|---|
| Point-Slope | y − 2.0000 = 2.0000(x − 1.0000) | When you know a point and the slope |
| Slope-Intercept | y = 2.0000x | For graphing, reading slope & y-intercept |
| Standard Form | 2x + -1y = 0 | For systems of equations, integer coefficients |
Line Visualization (y values for x = −5 to 5)
x=-5
y=-10.00
x=-4
y=-8.00
x=-3
y=-6.00
x=-2
y=-4.00
x=-1
y=-2.00
x=0
y=0.00
x=1
y=2.00
x=2
y=4.00
x=3
y=6.00
x=4
y=8.00
x=5
y=10.00
Sample Points on the Line
| x | y |
|---|---|
| -5 | -10.0000 |
| -3 | -6.0000 |
| -1 | -2.0000 |
| 1 | 2.0000 |
| 3 | 6.0000 |
| 5 | 10.0000 |
Planning notes, formulas, and examples
About the Point-Slope Form Calculator
The point-slope form of a linear equation is one of the most versatile ways to express a straight line in coordinate geometry. Written as y − y₁ = m(x − x₁), it directly encodes a known point (x₁, y₁) and the slope m, making it the natural choice when you are given exactly those two pieces of information. From there you can rearrange the equation into the more familiar slope-intercept form y = mx + b for quick graphing, or into standard form Ax + By = C for solving systems of equations.
This calculator handles four common tasks on the same page. First, enter two points to find the slope and all three equation forms automatically. Second, supply a single point and a slope to generate the equation directly. Third, compute the equation of a line parallel to a reference line through a given point (same slope). Fourth, compute the perpendicular line (negative reciprocal slope). Along the way the calculator reports the x-intercept, the angle the line makes with the x-axis, and the Euclidean distance between your two points.
It is useful when you want to compare equation forms instead of stopping at the first one that works. The comparison table shows how the same line looks in point-slope, slope-intercept, and standard form, while the plot and value table make it easier to catch sign errors and vertical-line edge cases.
When This Page Helps
Converting between two-point, point-slope, slope-intercept, and standard form by hand involves multiple rearrangement steps where sign errors are easy to make. This calculator does all four conversions at once, so you can compare forms instead of trusting a single rearrangement.
It is also useful when the line equation is only one part of the problem. Parallel and perpendicular slopes, intercepts, angle, and the x-to-y value table all come from the same setup, so checking them together makes it easier to spot a wrong sign or a mis-entered point.
How to Use the Inputs
- Choose a mode: Two Points, Point & Slope, Parallel Line, or Perpendicular Line.
- Enter the coordinates of your known point (x₁, y₁).
- For "Two Points" mode, enter a second point (x₂, y₂).
- For "Point & Slope" mode, enter the slope m.
- For parallel/perpendicular, enter the reference line's slope.
- Read the slope, y-intercept, x-intercept, and angle outputs.
- Compare all three equation forms in the table below the outputs.
- Use the line visualization to see y values from x = −5 to 5.
Formula used
Point-Slope Form: y − y₁ = m(x − x₁)
Slope: m = (y₂ − y₁) / (x₂ − x₁)
Slope-Intercept: y = mx + b where b = y₁ − m·x₁
Standard Form: Ax + By = C (A, B, C integers, A ≥ 0)
Parallel slope = m, Perpendicular slope = −1/mExample Calculation
Result: y − 2 = 2(x − 1) → y = 2x → 2x − y = 0
Slope m = (8 − 2)/(4 − 1) = 6/3 = 2. Using point (1,2): y − 2 = 2(x − 1). Expanding: y = 2x + 0. Standard form: 2x − y = 0.
Tips & Best Practices
- Use two-point mode when you only have coordinates and need to find the slope.
- Parallel lines share the same slope; perpendicular lines have slopes whose product is −1.
- Standard form is ideal for systems of equations because coefficients are integers.
- A zero slope means a horizontal line; an undefined slope means a vertical line (x = constant).
- Check your work by plugging both original points back into the resulting equation.
Three Forms of a Line and When to Use Each
Point-slope form y − y₁ = m(x − x₁) is best when you know a point and a slope. Slope-intercept y = mx + b is ideal for graphing and reading off the y-intercept. Standard form Ax + By = C (with integer coefficients) is preferred for systems of equations because elimination is cleaner. This calculator converts between all three, so you enter whichever data you have and immediately see the other representations.
Parallel and Perpendicular Lines
Two lines are parallel when their slopes are equal and perpendicular when the product of their slopes is −1 (i.e., m⊥ = −1/m). The calculator shows both the parallel and perpendicular slopes alongside the original, and you can enter a reference slope to compare against. These relationships are fundamental in analytic geometry, CAD, and collision detection.
Special Cases — Horizontal and Vertical Lines
A horizontal line has slope 0 (equation y = b); a vertical line has undefined slope (equation x = a). Point-slope form cannot represent vertical lines, which is why the calculator flags this case and falls back to the x = constant form. When the two input points share the same x-coordinate, the slope division produces infinity — the calculator detects this and reports the vertical line rather than crashing.
Sources & Methodology
Last updated:
Frequently Asked Questions
It is the equation y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any known point on the line.
Distribute m on the right side and add y₁ to both sides to get y = mx + b.
Standard form Ax + By = C is preferred when solving systems of linear equations or when integer coefficients are required.
The line is vertical (x = constant), and the slope is undefined. Point-slope form does not apply to vertical lines.
Parallel lines have equal slopes. Perpendicular lines have slopes whose product is −1, meaning the perpendicular slope is −1/m.
No — point-slope form applies only to 2D lines. For 3D, parametric or vector form is used instead.
It is the angle (in degrees) between the line and the positive x-axis, calculated as arctan(m).
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