How to solve linear equations, step by step

What it means to solve an equation

An equation says two expressions are equal. Solving it means finding the value of the variable that makes that statement true. For 3x + 7 = 22, the answer x = 5 works because 3 times 5, plus 7, is 22.

The picture to keep in mind is a balance scale. Whatever you do to one side, you must do to the other, or it tips. Adding, subtracting, multiplying, or dividing both sides by the same number (never dividing by zero) keeps the balance, and each move gives a simpler equation with the same solution.

A linear equation is one where the variable appears only to the first power: no x squared, and no x in a denominator. Most linear equations have exactly one solution; the two exceptions are covered below.

The general method

The same few moves solve almost every linear equation you will meet in middle school and Algebra I. Not every equation needs every step, but the order below avoids most trouble:

• Simplify each side: distribute to clear parentheses and combine like terms.
• If there are fractions, multiply both sides by a common denominator to clear them.
• Move the variable terms to one side by adding or subtracting.
• Move the constant terms to the other side the same way.
• Divide both sides by the coefficient of the variable.
• Check by substituting the answer into the original equation.

Worked example: a two-step equation

Start with an equation that needs two moves. Undo the addition first, then the multiplication, which is the reverse of the order of operations.

Worked example: parentheses and variables on both sides

When the variable appears on both sides, gather it on one side first. Subtracting the smaller variable term keeps the coefficient positive, which avoids sign errors.

Worked example: an equation with fractions

Fractions are where most mistakes happen, so clear them first by multiplying every term on both sides by the least common denominator.

When there is no solution, or every number works

Sometimes the variable disappears. 2(x + 3) = 2x + 6 simplifies to 2x + 6 = 2x + 6, which is true for every value of x, so the equation has infinitely many solutions. 2(x + 3) = 2x + 5 becomes 6 = 5 after subtracting 2x from both sides, which is never true, so there is no solution.

Neither result is a mistake. They are answers, and recognizing them is part of the grade 8 standards in California.

How equation solving builds in California

Under the California Common Core State Standards for Mathematics, solving equations grows steadily from grade 6 through Algebra I:

• Grade 6: one-step equations such as x + 9 = 15 or 4x = 28.
• Grade 7: two-step equations of the forms px + q = r and p(x + q) = r, often written from word problems.
• Grade 8: equations with rational coefficients and variables on both sides, distributing and combining like terms, and telling whether an equation has one solution, infinitely many, or none.
• Algebra I: explaining each step as a reasoned move, solving linear equations and inequalities, including ones with letters as coefficients, and writing equations from situations.

Common mistakes and how to avoid them

Most errors in equation solving are small slips that a quick check would catch. These come up most often:

• Distributing to only the first term, so 5(x − 2) becomes 5x − 2 instead of 5x − 10.
• Doing an operation to one side only, often while moving a term across the equals sign.
• Losing a negative sign when subtracting a negative number or a term with a minus in front of it.
• Multiplying only some of the terms by the common denominator when clearing fractions.
• Skipping the check. Substituting into the original equation takes seconds and catches almost everything above.

Practicing so it sticks

Say each step out loud before you write it: what is being done to x, and how do you undo it? Writing the full step, such as subtracting 7 from both sides, instead of moving numbers across the equals sign, slows you down just enough to keep signs straight while you are learning.