Linear Equation Solver – Solve ax + b = c Online
Solve any linear equation of the form ax + b = c instantly with our free online linear equation solver. Get step-by-step solutions for one-variable linear equations.
Equation Format:
ax + b = c
Understanding Linear Equations
A linear equation in one variable is an equation where the variable (usually x) appears only to the first power – no x², no x³, just plain x. The standard form is ax + b = c, where a, b, and c are known numbers and a ≠ 0.
Solving a linear equation means finding the value of x that makes the equation true. There's always exactly one solution (unless a = 0, which creates special cases). The process involves isolating x on one side of the equation.
Solving Method
- 1
Identify the coefficients
Write your equation in the form ax + b = c. Identify which numbers are a, b, and c.
- 2
Subtract b from both sides
This moves the constant term to the right side: ax = c - b
- 3
Divide both sides by a
This isolates x: x = (c - b) / a
- 4
Verify your answer
Plug your solution back into the original equation to confirm it works.
Worked Examples
Example 1: Basic Equation
Solve: 2x + 5 = 15
Example 2: Negative Coefficient
Solve: -4x + 12 = 28
Example 3: Negative Constant
Solve: 3x - 7 = 14
Example 4: No x Term on Right
Solve: 7x - 21 = 0
Quick Fact
The word "algebra" comes from the Arabic "al-jabr," meaning "reunion of broken parts." It was coined by the 9th-century Persian mathematician Al-Khwarizmi in his book about solving linear and quadratic equations. His methods for balancing equations are still taught today.
Frequently Asked Questions
What if a = 0?
If a = 0, the equation becomes b = c. If b equals c, any value of x works (infinite solutions). If b doesn't equal c, no value of x works (no solution). These are called degenerate cases.
Can the solution be a fraction or decimal?
Absolutely. If (c - b) doesn't divide evenly by a, you'll get a fraction or decimal. For example, 2x = 7 gives x = 3.5. Both are perfectly valid solutions.
What if my equation isn't in ax + b = c form?
Rearrange it first. Move all x terms to one side and constants to the other. For example, 3x + 5 = 2x + 10 becomes 3x - 2x = 10 - 5, which simplifies to x = 5.
Why do I need to do the same thing to both sides?
An equation is a balance – both sides are equal. If you change one side without changing the other the same way, you break the equality. Doing the same operation to both sides preserves the balance.
How do I check my answer?
Substitute your solution back into the original equation. If both sides equal the same number, your answer is correct. Always check – it catches careless errors.
What's the difference between linear and quadratic equations?
Linear equations have x to the first power only (ax + b = c) and always have one solution. Quadratic equations have x² terms (ax² + bx + c = 0) and can have zero, one, or two solutions.
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