How to Solve Linear Equations Step by Step

What a linear equation is

A linear equation in one variable is an equation that can be rearranged into the form $ax + b = 0$, where $a$ and $b$ are numbers and $a \neq 0$. The unknown appears only to the first power: there is no $x^2$, no $\sqrt{x}$, and no $x$ in a denominator. Because the graph of $y = ax + b$ is a straight line, the equation has exactly one solution, $x = -\frac{b}{a}$, which is where that line crosses the x-axis.

Equations such as $3x + 7 = 22$, $5(x - 2) = 3x + 4$ and $\frac{x}{3} + \frac{x}{4} = 7$ are all linear, even though they do not look like $ax + b = 0$ yet. The job is to rewrite them, one legal step at a time, until the unknown stands alone.

The balance rule behind every step

Think of an equation as a balanced scale. You may do anything you like to one side as long as you do exactly the same thing to the other side, because equal quantities changed in the same way stay equal. The four operations you will use are:

• add the same number or expression to both sides;
• subtract the same number or expression from both sides;
• multiply both sides by the same non-zero number;
• divide both sides by the same non-zero number.

Multiplying or dividing by zero is the one forbidden move: multiplying by zero turns any equation into $0 = 0$ and destroys the information, and dividing by zero is undefined.

A five-step routine

1. Clear grouping symbols by distributing: $5(x - 2)$ becomes $5x - 10$.
2. Clear fractions or decimals by multiplying both sides by the least common denominator or by a power of 10.
3. Combine like terms on each side separately.
4. Collect the variable terms on one side and the constant terms on the other by adding or subtracting.
5. Divide by the coefficient of the variable, then substitute the answer back into the original equation to check it.

Not every equation needs every step. The routine simply gives you an order that avoids the most common tangles, such as trying to divide before like terms have been combined.

Worked examples from simple to harder

Notice the order. Seven was added last when the expression $3x + 7$ was built, so it is removed first. Solving reverses the order of operations.

If you prefer a positive coefficient, collect the variable terms on the side that already has the larger coefficient. In Example 3, adding $4x$ to both sides gives $9 = 6x - 15$, then $24 = 6x$ and $x = 4$, with no negative division at all.

Equations with fractions and decimals

Fractions are where most errors happen, so remove them at the start. Multiply every term on both sides by the least common denominator (LCD). Every term means every term, including whole numbers that have no visible denominator.

Decimals work the same way with a power of ten. Choose the power that clears the decimal with the most places.

You can confirm fraction-heavy equations with the fraction calculator or compare each step with the equation solver.

No solution and infinitely many solutions

Sometimes the variable disappears completely while you are solving. What is left tells you which special case you have.

A true statement such as $6 = 6$ means the two sides are the same expression, so every real number is a solution. The equation is called an identity.

A false statement such as $6 = 5$ means no value of $x$ can ever make the sides equal: the equation has no solution. Graphically, the two sides are parallel lines with the same slope and different intercepts, so they never meet.

Solving for one variable in a formula

The same balance rule rearranges formulas, which is often called solving a literal equation. Treat every letter except the one you want as if it were a number.

Rewriting an equation this way is the first step for graphing it, which you can explore with the slope calculator.

Turning a word problem into an equation

Word problems become linear equations when one quantity changes at a constant rate. Name the unknown, write each phrase as an expression, and set the two descriptions of the same amount equal.

More practice translating sentences into equations is in the word problem solver.

Common mistakes and how to avoid them

• Distributing to the first term only: $3(x + 4)$ is $3x + 12$, not $3x + 4$.
• Losing a negative sign: $-(x - 5)$ is $-x + 5$. Distribute the minus to every term in the bracket.
• Multiplying only the fractions by the LCD and forgetting the whole-number terms.
• Moving a term across the equals sign without changing its sign. Saying a term moves is shorthand for subtracting it from both sides.
• Stopping at $-x = 3$. Divide by $-1$ to get $x = -3$.
• Skipping the check. Substituting into the original equation, not a later line, catches errors made in any step.

Where to go next

Once linear equations feel automatic, the same skills carry straight into linear inequalities, where one extra rule applies, and into systems of equations, where two linear equations share two unknowns. If an equation contains $x^2$, read the guide to solving quadratic equations. To practise, type any of the examples above into the solve for x calculator and compare its steps with yours.

Frequently asked questions