Solving Linear Equations

A plumber charges $45 upfront plus $30 per hour. The bill comes to $165. How many hours did the job take? That question is a linear equation: 30x + 45 = 165. Solving it means finding the value of x that makes both sides equal — and the answer is x = 4 hours. Linear equations are any equation where the variable appears to the first power, and they are the foundation of every algebra course. They scale from one-step problems to multi-step expressions with fractions, decimals, and variables on both sides, and they are the core skill behind systems of equations, graphing lines, and functions.

1) What a Linear Equation Is

A linear equation in one variable has the form ax + b = c, where a, b, and c are constants and a is not zero. The variable x appears only to the first power. Examples: 3x + 5 = 14, 2(x − 3) = 8, and x/4 + 1 = 7 are all linear equations. An equation like x² + 3 = 7 is not linear — it is quadratic.

The solution to a linear equation is the value of x that makes both sides equal. Most linear equations have exactly one solution, but two special cases produce either no solution or infinitely many solutions. The flowchart below shows the full solving process and when to expect each outcome.

Linear equations are the building block for the rest of algebra. Once you can solve them confidently, systems of equations and graphing lines follow directly.

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2) One-Step Equations

A one-step equation requires a single operation to isolate the variable. The operation you apply is always the inverse of what is being done to x.

Addition and subtraction

If a number is being added to x, subtract it from both sides. If a number is being subtracted from x, add it to both sides.

Multiplication and division

If x is being multiplied by a coefficient, divide both sides by that coefficient. If x is being divided, multiply both sides by the divisor.

Negative coefficients

When the coefficient is negative, dividing by a negative number flips the sign of the answer. This is the most common sign error in one-step equations.

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3) Multi-Step Equations

Multi-step equations require two or more operations. The strategy is to undo operations in reverse order — the opposite of the order of operations. Addition and subtraction are undone first, then multiplication and division. Before any of that, simplify: distribute parentheses and combine like terms on each side.

Standard multi-step example

With the distributive property

When parentheses are present, distribute before doing anything else. Skipping distribution and moving the constant first is one of the most common multi-step errors.

Combining like terms first

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4) Variables on Both Sides

When x appears on both sides of the equation, the first move is to collect all variable terms on one side. It does not matter which side you choose — pick the side that keeps the coefficient positive to reduce sign errors.

With distribution on both sides

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5) Equations with Fractions and Decimals

Fractions and decimals in linear equations are not harder to solve — they just require one extra step at the start. Clearing fractions or decimals converts the equation into a standard integer equation before you begin solving.

Clearing fractions using the LCD

Find the least common denominator (LCD) of all fractions in the equation. Multiply every term on both sides by the LCD. Every fraction cancels, leaving an integer equation.

Multi-step fraction example

Clearing decimals

Identify the decimal with the most decimal places. Multiply every term by the corresponding power of 10 to eliminate all decimals.

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6) Special Cases: No Solution and All Real Numbers

Most linear equations have exactly one solution. But when variable terms on both sides have the same coefficient, they cancel during solving and leave a statement with no variable. What that statement says determines the outcome.

No solution — worked example

All real numbers — worked example

How to write these answers on homework platforms

For no solution: MyMathLab accepts “No solution” or the empty set symbol. ALEKS accepts “No solution.” WebAssign typically accepts “DNE” or “no solution.” For all real numbers: MyMathLab and ALEKS typically accept “All real numbers” or the interval notation (−∞, ∞). Check your specific course settings — accepted formats vary by version and instructor configuration.

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7) Word Problems Using Linear Equations

Word problems are how linear equations are tested on most algebra platforms. The algebra is the same as in the previous sections — the skill being tested is translating the scenario into an equation. Every word problem that produces a single linear equation follows the same setup: identify the unknown, assign a variable, write an equation from the relationships in the problem, solve.

Type 1: Rate and cost problems

These problems describe a fixed charge plus a per-unit rate. Set the total equal to the expression and solve for the unknown quantity.

Type 2: Consecutive integer problems

Consecutive integers differ by 1. Consecutive even or odd integers differ by 2. Assign the first integer as x, write the others in terms of x, set their sum or relationship equal to the given total.

Type 3: Percent and discount problems

Percent problems translate directly into multiplication. “What is 20% of x” becomes 0.20x. “A price after a 15% discount” becomes 0.85x (the original minus 15% of it).

Type 4: Geometry problems

Perimeter problems, angle relationships, and area problems with one unknown all produce linear equations. Write the geometric formula, substitute the expressions given, and solve.

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8) Where Students Lose Points

Not distributing before moving terms

Sign errors when subtracting a negative

Dividing only the last term by the coefficient

Not applying the operation to the entire opposite side

When you add or subtract a term, it must be added or subtracted from the entire other side, not just one term. Write the operation below both sides explicitly until it is automatic.

Stopping at one solution when the variable cancelled

If you reach a statement like 8 = 8, you have not made an error — this is a valid outcome meaning all real numbers. Students sometimes assume they made a mistake and restart. If you reach 8 = 3, same thing — the answer is no solution, not an arithmetic error to fix.

Fraction entry format on platforms

ALEKS requires fractions to be fully reduced. MyMathLab uses a fraction entry tool — typing 3/5 as plain text may not register correctly. Use the answer entry palette. For detailed format guidance: MyMathLab Help.

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9) How FMMC Can Help

Linear equations appear in every algebra course and on every major homework platform. If you are behind on assignments or facing a proctored exam on this material, FMMC’s algebra experts handle every module with an A/B guarantee.

Other guides in this series

Solving linear equations feeds directly into systems of equations — every substitution and elimination step relies on these same mechanics — and into graphing lines, where slope-intercept form requires isolating y.

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FAQ: Solving Linear Equations

What is a linear equation?

A linear equation is an equation where the variable appears to the first power only — no exponents greater than 1, no square roots. It has the general form ax + b = c. The solution is the value of x that makes both sides equal.

What is the first step in solving a linear equation?

Simplify each side first. Distribute any parentheses and combine like terms on each side before moving any terms across the equals sign. Skipping simplification is the most common source of errors in multi-step equations.

How do you solve an equation with variables on both sides?

Move all variable terms to one side by adding or subtracting. Choose the side that gives a positive coefficient to reduce sign errors. Then move all constant terms to the other side and divide by the coefficient. If the variable terms cancel completely, check whether the remaining statement is true or false to determine if the solution is all real numbers or no solution.

How do you solve a linear equation with fractions?

Find the LCD of all fractions in the equation and multiply every term on both sides by that LCD. This clears all fractions and leaves a standard integer equation. Then solve normally. Always check the solution by substituting back into the original equation with fractions.

What does it mean when you get a false statement like 5 = 9?

It means the equation has no solution. The variable terms cancelled out and left a contradiction that is never true. There is no value of x that satisfies the equation. This is not an arithmetic error — it is a valid outcome. Write “No solution” or the empty set symbol as your answer.

What does it mean when you get a true statement like 5 = 5?

It means every value of x satisfies the equation. The solution is all real numbers. The two sides of the original equation are identical expressions, so no matter what x equals, both sides will be equal. Write “All real numbers” or the interval (−∞, ∞) as your answer.

How do you check a solution to a linear equation?

Substitute the value you found back into the original equation and simplify both sides. If both sides equal the same number, the solution is correct. Always check in the original equation, not a simplified version — an error in an earlier step may have simplified into something that looks correct but is not.

Can FMMC help with linear equations homework and exams?

Yes. FMMC handles algebra homework, quizzes, and proctored exams across MyMathLab, ALEKS, WebAssign, and Hawkes Learning. See our algebra homework help page or get a free quote.

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