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.
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 in one tool. 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.
Whether you are a student practicing algebra homework, a teacher preparing examples, or an engineer verifying coordinate geometry, this tool eliminates arithmetic errors and produces a clean comparison table of all three standard equation forms so you can pick the version that best suits your problem. Presets let you explore common lines instantly, and the visual bar chart shows how y values change from x = −5 to 5.
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, showing every algebraic step so you can follow along. It also computes the parallel and perpendicular slopes, the x- and y-intercepts, the line angle, and a value table from x = −5 to 5 — giving you a complete picture of the line from a single pair of inputs. Students use it to check homework; engineers use it for quick linear-interpolation references.
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/m
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.
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.
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.
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.
It is the equation y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any known point on the line. Use this as a practical reminder before finalizing the result.
Distribute m on the right side and add y₁ to both sides to get y = mx + b. Keep this note short and outcome-focused for reuse.
Standard form Ax + By = C is preferred when solving systems of linear equations or when integer coefficients are required. Apply this check where your workflow is most sensitive.
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). Use this checkpoint when values look unexpected.