Linear Equation Solver
Solve and graph linear equations in any form — step-by-step
Equation Preview
y = 2x − 1
Enter slope (m) and y-intercept (b) for y = mx + b
Rise over run
Where line crosses y-axis
Optional: evaluate at a specific x or y value
Quick Examples
Enter Your Equation
Choose a mode above, fill in the coefficients, and your solution with step-by-step working and a graph will appear here instantly.
How to Use the Linear Equation Solver
Choose Your Equation Form
Select the mode that matches your equation: Single Variable (ax + b = c) to find one unknown x, Slope-Intercept to work with y = mx + b, Two Points to find the line through two coordinates, or Standard Form to convert Ax + By = C. The input fields update instantly when you switch modes.
Enter Your Coefficients
Type the numeric values into the fields. Decimals and negative numbers are fully supported — enter them with a minus sign (e.g., −3.5). Use the Quick Examples buttons to load a sample equation immediately. The equation preview at the top updates in real time as you type so you can confirm you have entered it correctly.
Review the Solution and Graph
Results appear automatically as you type. The hero section shows the primary answer. Below it, all three equation forms (slope-intercept, standard, point-slope) are shown side by side. Key properties — slope, intercepts, angle of inclination, and parallel/perpendicular slopes — are listed below. The line graph plots the equation from x = −10 to x = 10 with intercepts marked.
Export or Print Your Results
Click Copy Results to copy all key values to your clipboard for pasting into notes or a homework document. Click Export CSV to download a spreadsheet-ready file. Click Print to open the print dialog for a clean printout. Use the Step-by-Step Solution panel to expand the full arithmetic working — ideal for checking your own work or understanding the method.
Frequently Asked Questions
What is the difference between slope-intercept form and standard form?
Slope-intercept form (y = mx + b) is the most common in algebra courses because the slope m and y-intercept b are immediately visible without any manipulation. It is ideal when you want to graph the line quickly. Standard form (Ax + By = C) uses integer coefficients and is preferred in some textbooks, standardized tests, and computer algorithms because it treats x and y symmetrically. Both forms represent exactly the same line — this solver converts between them automatically so you always have both. Point-slope form (y − y₁ = m(x − x₁)) is most useful when you know one point and the slope but not the y-intercept.
How do I find the slope of a line given two points?
The slope formula is m = (y₂ − y₁) / (x₂ − x₁), often described as 'rise over run'. Rise is the vertical change (y₂ − y₁) and run is the horizontal change (x₂ − x₁). A positive slope means the line goes up from left to right. A negative slope means it goes down. A slope of zero means the line is perfectly horizontal. An undefined slope (division by zero) means the line is vertical — x₁ = x₂. Enter both points in Two Points mode and the solver calculates the slope, y-intercept, line equation, distance, and midpoint automatically.
What does the angle of inclination mean?
The angle of inclination (θ) is the angle the line makes with the positive x-axis, measured counterclockwise. It is calculated as θ = arctan(m) × (180/π). A horizontal line has θ = 0°. A line with slope 1 has θ = 45°. A line with slope −1 has θ = −45° (or 135° measured from the positive x-axis). Vertical lines have θ = 90° and an undefined slope. The angle is useful in trigonometry, navigation, and any application where direction or bearing matters, such as calculating the angle of a ramp or the elevation of a hill.
What is the perpendicular slope and when do I need it?
Two lines are perpendicular if they cross at exactly a 90° angle. The slopes of perpendicular lines satisfy m₁ × m₂ = −1, so the perpendicular slope is −1/m. For example, if a line has slope 3, the perpendicular slope is −1/3. If a line has slope −2, the perpendicular is 1/2. You need this when constructing a normal to a surface, finding the shortest distance from a point to a line, building right angles in geometry proofs, or in computer graphics when calculating reflections and shadows. The solver displays the perpendicular slope alongside the parallel slope in every result.
How do I convert standard form to slope-intercept form?
Starting from Ax + By = C, isolate y by subtracting Ax from both sides to get By = −Ax + C, then divide every term by B to get y = (−A/B)x + C/B. The slope is m = −A/B and the y-intercept is b = C/B. For example, 3x − 2y = 6 becomes −2y = −3x + 6, then y = (3/2)x − 3, so m = 1.5 and b = −3. If B = 0, the equation represents a vertical line x = C/A with an undefined slope. This solver performs the conversion automatically in Standard Form mode.
Can this solver handle decimal and fraction coefficients?
Yes. All input fields accept any real number including decimals (e.g., 1.5, −0.75, 3.14159) and integers. There is no fraction input mode, but you can enter decimal equivalents of fractions: 1/2 = 0.5, 1/3 ≈ 0.3333, 2/5 = 0.4. The decimal precision control lets you choose 2, 4, 6, or 8 decimal places in the output. Results are automatically rounded near integer values to avoid floating-point noise (e.g., 2.9999999 is displayed as 3). For exact fraction arithmetic, a Computer Algebra System (CAS) like Wolfram Alpha would be more appropriate.