📐 Algebra Solver

Linear Equation Solver (ax + b = c)

Solve single-variable linear equations instantly with clear step-by-step solutions using the balance method.

Linear Equation

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Enter Linear Equation to Solve ℹ️ Info

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How to Use & Educational Guide

In-depth guide for Linear Equation Solver (ax + b = c)

Educational Overview

What Is a Linear Equation in One Variable?

A linear equation in one variable is an equation in which the highest power of the variable is 1. A common form is $ax + b = c$, where $x$ is the unknown and $a$, $b$, and $c$ are constants. Solving the equation means finding the value of $x$ that makes both sides equal.

Linear Equation Formula

For a linear equation in the form $ax + b = c$, isolate the variable by using inverse operations:

$$ax + b = c \implies ax = c - b \implies x = \frac{c-b}{a} \quad (a \ne 0)$$

Here, a is the coefficient of $x$, b and c are constants, and x is the unknown variable.

How to Solve a Linear Equation Step-by-Step?

A linear equation solver with steps uses inverse operations to isolate the unknown variable. The same operation must be applied to both sides of the equation to maintain equality.

Step 1 (Remove the Constant): Add or subtract the constant from both sides to isolate the variable term. For $ax + b = c$, subtract $b$ from both sides.
Step 2 (Isolate the Variable): Divide both sides by the coefficient of $x$ to leave the variable by itself.
Step 3 (Simplify the Answer): Calculate the resulting fraction or decimal to obtain the value of $x$.
Step 4 (Check the Solution): Substitute the calculated value back into the original equation and verify that both sides have the same value.

Linear Equation Solver Examples

Example 1: Solve $2x + 4 = 10$

• Step 1: Subtract 4 from both sides: $2x = 10 - 4 = 6$.

• Step 2: Divide both sides by 2: $x = 6 \div 2 = 3$.

• Step 3 (Check): $2(3) + 4 = 10$.

👉 Solution: $x = 3$

Example 2: Solve $-3x - 7 = 14$

• Step 1: Add 7 to both sides: $-3x = 14 + 7 = 21$.

• Step 2: Divide by $-3$: $x = 21 \div (-3) = -7$.

• Step 3 (Check): $-3(-7) - 7 = 21 - 7 = 14$.

👉 Solution: $x = -7$

Example 3: Solve $0.5x + 1.2 = 3.7$

• Step 1: Subtract 1.2 from both sides: $0.5x = 2.5$.

• Step 2: Divide by 0.5: $x = 2.5 \div 0.5 = 5$.

• Step 3 (Check): $0.5(5) + 1.2 = 3.7$.

👉 Solution: $x = 5$

Example 4: Solve $\frac{x}{3} + 4 = 9$

• Step 1: Subtract 4 from both sides: $\frac{x}{3} = 5$.

• Step 2: Multiply both sides by 3: $x = 15$.

• Step 3 (Check): $\frac{15}{3} + 4 = 5 + 4 = 9$.

👉 Solution: $x = 15$

Single-Variable vs. System of Linear Equations

➗ One-Variable Equation

This calculator solves equations containing one unknown, such as $2x + 4 = 10$. The goal is to find the value of $x$.

🔢 Systems of Equations

A system of linear equations contains two or more equations and may involve multiple unknowns. These require methods such as substitution, elimination, or matrix methods.

Where Are Linear Equations Used?

💼 Business & Costs

Businesses use linear equations to calculate unknown costs, quantities, revenue, and break-even points. For example, $C = ax + b$ can represent total cost.

🚗 Distance & Motion

Linear relationships can help find an unknown time, distance, or rate when motion occurs at a constant speed.

Common Mistakes to Avoid

Changing Only One Side: Any addition, subtraction, multiplication, or division must be applied to both sides of the equation.
Sign Errors: Pay close attention when adding or subtracting negative numbers. For example, $-3x = 21$ gives $x = -7$.
Forgetting the Coefficient: When $x$ appears alone, its coefficient is 1. Therefore, $x = 1x$.
Not Checking the Answer: Substitute the final value back into the original equation to confirm that the equation is true.

Can This Calculator Solve Multiple Variables?

This online linear equation solver is designed for single-variable equations in the form $ax + b = c$. Equations with two unknowns, three or four variables, matrices, or multiple linear equations form a different problem called a system of linear equations. Non-linear equations, where variables have powers other than 1, also require different solving methods.

Mathematical accuracy verified by AptCalc Engine

Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What is a linear equation?

A linear equation is an equation in which the highest power of the variable is 1. A common one-variable form is ax + b = c, where a is not zero.

How do you solve a linear equation step by step?

First isolate the variable term by adding or subtracting the constant from both sides. Then divide by the coefficient of the variable and check the result in the original equation.

What does a linear equation solver calculate?

A linear equation solver finds the unknown value that makes a linear equation true. This calculator solves single-variable equations such as 2x + 4 = 10.

Can a linear equation solver solve 2 unknowns?

A single-variable linear equation solver cannot determine two unknowns from one equation. Two or more unknowns generally require a system of linear equations with enough independent equations.

What is the difference between linear and non-linear equations?

A linear equation has variables only to the first power, such as 2x + 4 = 10. A non-linear equation contains powers, products, roots, or other forms that make the relationship non-linear.