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    Linear Equation Calculator

    Solve linear equations of the form ax + b = c

    Enter values for: ax + b = c

    How It Works

    Overview

    A linear equation in one variable has the standard form ax + b = c, where a, b, and c are constants and a ≠ 0. "Linear" means the variable appears only to the first power — no x², no √x, no 1/x — so the graph of y = ax + b is a straight line.

    Solving a linear equation means isolating x on one side using inverse operations. The two-step recipe is: subtract b from both sides, then divide both sides by a. The result is a single value of x that makes both sides of the equation equal — the equation's unique solution.

    The Formula

    x = (c − b) / a

    Derived from ax + b = c by isolating x:

    • Start: ax + b = c
    • Subtract b from both sides: ax = c − b
    • Divide both sides by a: x = (c − b) / a

    This is closely related to slope-intercept form y = mx + b, which describes the same kind of relationship as a line in 2D: m is the slope (the coefficient of x) and b is the y-intercept (where the line crosses the y-axis).

    Worked Example

    Solve 3x + 7 = 22. Here a = 3, b = 7, c = 22.

    • Subtract 7 from both sides: 3x = 22 − 7 = 15
    • Divide by 3: x = 15 / 3 = 5
    • Check: 3(5) + 7 = 15 + 7 = 22 ✓

    A real-world example: a phone plan charges a $15 monthly fee plus $0.10 per minute. Your bill was $42.50 — how many minutes did you use? Set up the equation 0.10·m + 15 = 42.50, giving m = (42.50 − 15)/0.10 = 275 minutes.

    When to Use This

    • Algebra homework — the most common type of equation in early algebra courses.
    • Cost and revenue problems — fixed cost plus per-unit cost equals total budget.
    • Unit conversions with offsets, like Fahrenheit → Celsius: F = (9/5)C + 32, solved for C given F.
    • Break-even analysis — find the production volume where revenue equals total cost.
    • Recipe scaling, motion problems (distance = rate × time + offset), and any "rate plus a flat amount" scenario.

    Common Mistakes to Avoid

    • Doing the operation to only one side. Whatever you do — add, subtract, multiply, divide — must be applied to both sides to keep the equation balanced.
    • Sign mistakes when moving terms. Moving +7 to the other side makes it −7, not +7.
    • Dividing before subtracting. Order matters: subtract b first, then divide by a.
    • Distributing incorrectly. 3(x + 2) = 3x + 6, not 3x + 2. Always multiply through every term inside the parentheses.
    • Forgetting to verify. Plug your answer back into the original equation — it's the fastest way to catch arithmetic errors.

    Frequently Asked Questions

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