Linear Equation Solver

Solve linear equations in one variable with detailed step-by-step working. Enter equations in the form ax + b = c or ax + b = cx + d and get the exact solution for x with a verification check.

2x + 3 = 7

a (coefficient of x)

b (constant term, left side)

c (right-hand value)

Solution for x
x = 2
Exact solution

Equation

2x + 3 = 7

Linear equation in one variable

📝 Step-by-Step Solution

✅ Verified: 2(2) + 3 = 7 ✓

✅ Equation solved successfully!

❌ Please check your equation values.

📐 Real-World Linear Equation Examples

📘 Simple Equation: 2x + 3 = 7

Solve for x in the equation 2x + 3 = 7.

Step 1: Subtract 3 from both sides: 2x = 7 − 3 = 4

Step 2: Divide both sides by 2: x = 4 ÷ 2 = 2

Verification: 2(2) + 3 = 4 + 3 = 7 ✓

This is the most common form of linear equation encountered in algebra.

📗 Equation with Both Sides: 3x + 5 = x + 9

Solve for x when x appears on both sides: 3x + 5 = x + 9.

Step 1: Subtract x from both sides: 2x + 5 = 9

Step 2: Subtract 5 from both sides: 2x = 4

Step 3: Divide both sides by 2: x = 2

Verification: 3(2) + 5 = 11 and 2 + 9 = 11 ✓

📕 Equation with Negative Coefficients: 4x − 8 = 12

Solve for x: 4x − 8 = 12.

Step 1: Add 8 to both sides: 4x = 12 + 8 = 20

Step 2: Divide both sides by 4: x = 20 ÷ 4 = 5

Verification: 4(5) − 8 = 20 − 8 = 12 ✓

📙 Real-World Application: Budget Problem

You have $50 for a month of music streaming. The service costs $8 per month plus a $10 sign-up fee. How many months can you subscribe?

Equation: 8x + 10 = 50, where x = number of months

Step 1: Subtract 10: 8x = 40

Step 2: Divide by 8: x = 5 months

Verification: 8(5) + 10 = 40 + 10 = 50 ✓ You can subscribe for 5 months.

🔢 Understanding Linear Equations

A linear equation is an equation that represents a straight line when graphed. In one variable, a linear equation has the form ax + b = c, where a, b, and c are constants and a ≠ 0. The goal is to find the value of x that makes the equation true.

ax + b = c → x = (c − b) / a

Standard formula for solving linear equations in one variable, where a ≠ 0.

ax + b = cx + d → x = (d − b) / (a − c)

General formula when x appears on both sides of the equation, where a ≠ c.

How to Solve Step by Step

Identify the form: Determine if your equation is ax + b = c (simple) or ax + b = cx + d (x on both sides).

Collect variable terms: If x appears on both sides, subtract cx from both sides to get all x terms on the left: (a − c)x + b = d.

Isolate the variable term: Subtract the constant b from both sides to get (a − c)x = d − b (or ax = c − b for the simple form).

Solve for x: Divide both sides by the coefficient of x: x = (d − b) / (a − c) or x = (c − b) / a.

Verify: Substitute your answer back into the original equation to check it's correct.

Quick Tips for Solving

✅ Always Check Your Answer

Plug your solution back into the original equation. If both sides are equal, your answer is correct. This is the simplest way to catch mistakes.

🔄 Do the Same to Both Sides

Whatever operation you perform on one side of the equation, you must perform on the other side. This keeps the equation balanced.

⚠️ Watch for Special Cases

If a = 0 in ax + b = c, the equation is either impossible (b ≠ c) or always true (b = c). If a = c in the both-sides form, the x terms cancel and you may have a contradiction or identity.

📏 Keep Fractions as Fractions

When the solution is a fraction, leave it in fraction form for exact precision. Our calculator shows both exact fractional form and decimal approximation.

✨ Linear Equation Solver Features

🧮

Dual Equation Modes

Supports both ax + b = c and ax + b = cx + d equation formats. Toggle between modes seamlessly.

📝

Step-by-Step Working

Every calculation is broken down into clear, numbered steps showing each algebraic manipulation.

✅

Answer Verification

Automatically verifies the solution by substituting x back into the original equation to confirm correctness.

🔢

Exact & Decimal Results

Shows both exact fractional solutions and decimal approximations for complete precision.

📱

Mobile Friendly

Fully responsive design that works seamlessly on smartphones, tablets, and desktop computers.

🔒

Privacy Protected

All calculations are performed locally in your browser. Your data never leaves your device.

About the Linear Equation Solver

Our Linear Equation Solver is a comprehensive tool designed to help students, teachers, and professionals solve linear equations in one variable with complete transparency. Whether you're studying algebra, preparing for exams, or just need a quick calculation, this solver provides not just the answer but the complete step-by-step reasoning behind it.

The Formula Behind the Solver

A linear equation in one variable has the general form ax + b = c, where a, b, and c are constants and a ≠ 0. The solution is given by isolating x through inverse operations:

x = (c − b) / a

This is derived by subtracting b from both sides, then dividing by a.

For equations where x appears on both sides (ax + b = cx + d), the solution is:

x = (d − b) / (a − c)

Derived by subtracting cx and b from both sides, then dividing by (a − c).

Why Learn to Solve Linear Equations?

Linear equations are the foundation of algebra and appear in countless real-world applications:

Budgeting: Calculate how many months you can afford a subscription service: 8x + 10 = 50
Distance problems: If you drive at 60 mph, how long to cover 300 miles? 60t = 300
Temperature conversion: F = (9/5)C + 32 — a linear equation relating Fahrenheit and Celsius
Business: Break-even analysis where revenue equals cost
Science: Many physics formulas like v = u + at are linear equations

How to Use This Calculator

Step 1: Choose the Mode

Select between 'ax + b = c' (simple) and 'ax + b = cx + d' (x on both sides) using the toggle buttons at the top.

Step 2: Enter Values

Input the coefficients and constants. You can use whole numbers, decimals, or fractions. The equation preview updates in real time.

Step 3: Solve

Click "Solve for x" to see the solution, step-by-step working, and answer verification all at once.

Step 4: Verify

Check the verification section to confirm your answer is correct by seeing it substituted back into the original equation.

Tips for Solving Linear Equations

🧮 Work Backward Using Inverse Operations

To isolate x, think about what operations are applied to x and reverse them. If x is multiplied by a and then added to b, reverse by subtracting b first, then dividing by a. This "undoing" approach works for any linear equation.

📝 Keep the Equation Balanced

Whatever you do to one side, do to the other. This is the golden rule of algebra. Write each step clearly to avoid mistakes, especially with negative numbers and fractions.

🔍 Check Your Work

Always substitute your answer back into the original equation. Both sides should be equal. If they're not, go back and check each step — the mistake is often a simple arithmetic error.

Frequently Asked Questions (FAQ)

What is a linear equation?

A linear equation is an equation where the highest power of the variable (usually x) is 1. It forms a straight line when graphed. In one variable, it has the general form ax + b = c, where a, b, and c are constants and a ≠ 0. The graph of a linear equation in two variables (y = mx + b) is a straight line with slope m and y-intercept b.

How do I solve ax + b = c?

To solve ax + b = c for x: (1) Subtract b from both sides: ax = c − b. (2) Divide both sides by a: x = (c − b) / a. For example, 2x + 3 = 7 → 2x = 4 → x = 2. Always verify by plugging x back into the original equation.

How do I solve ax + b = cx + d?

To solve ax + b = cx + d: (1) Subtract cx from both sides: (a − c)x + b = d. (2) Subtract b from both sides: (a − c)x = d − b. (3) Divide by (a − c): x = (d − b) / (a − c). This works as long as a ≠ c, otherwise the x terms cancel.

What if a = 0 in my equation?

If a = 0, the equation 0x + b = c simplifies to b = c. If b equals c, the equation is an identity (true for all values of x). If b ≠ c, the equation is a contradiction (no solution). In the both-sides form, if a = c, the x terms cancel and you're left with b = d — again either an identity or contradiction.

What is the difference between linear and quadratic equations?

A linear equation has the variable raised to the power of 1 (x¹), while a quadratic equation has the variable raised to the power of 2 (x²). Linear equations have at most one solution, while quadratic equations can have up to two solutions. For example, 2x + 3 = 7 is linear (solution: x = 2), while x² − 5x + 6 = 0 is quadratic (solutions: x = 2, x = 3).

How can I check if my answer is correct?

The easiest way to check your answer is to substitute it back into the original equation. Replace every x with your solution and simplify both sides. If both sides are equal (e.g., 7 = 7), your answer is correct. Our calculator does this automatically and shows the verification step.

Can this calculator handle fractions and decimals?

Yes! Our Linear Equation Solver accepts whole numbers, decimals (like 0.5 or 3.14), and fractions (enter as decimals). The results are displayed as both simplified fractions and decimal approximations for maximum precision and clarity.

What real-world problems use linear equations?

Linear equations appear everywhere: calculating distance (distance = rate × time), converting temperatures (F = 9/5C + 32), determining costs (total cost = unit price × quantity + fixed fee), budgeting, calculating interest, mixing solutions, age problems, and many more. Any situation where one quantity varies at a constant rate with respect to another can be modeled with a linear equation.

What if the solution is a fraction?

Our calculator displays fractional solutions in simplified form. For example, if x = 3/4, we show "x = 3/4 (0.75)" — both the exact fraction and decimal approximation. Fractions are exact representations, while decimals may be rounded, so the fraction is the most precise answer.

Can I use this calculator for homework?

Absolutely! Our Linear Equation Solver is designed as a learning tool. It shows complete step-by-step working, helping you understand the solving process rather than just giving you the answer. Use it to check your work, see where you might have gone wrong, and build confidence in solving linear equations independently.

⚠️ Important Note: This Linear Equation Solver is for educational and informational purposes only. While the calculations are accurate, always double-check important results independently. The calculator assumes standard algebraic conventions and may not handle certain edge cases (such as symbolic variables or systems of equations). For complex mathematical problems, consult a qualified mathematics professional or instructor.