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Solving linear equations using multiplication and division

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Solving Equations with Multiplication and Division

A one-step linear equation with multiplication or division is solved using the inverse operation on both sides. If x is multiplied, divide both sides; if x is divided, multiply both sides. Learn the method with worked examples for 3x = 12 and x/4 = 5, plus how to check your answer.

One-step equations with multiplication and division

Some one-step linear equations have the variable being multiplied (like 3x = 12) or divided (like x/4 = 5), rather than added or subtracted — a case not covered by solving one-step equations with addition/subtraction. To solve either type, you undo the operation using its inverse — and the same operation must be applied to both sides of the equation to keep it balanced.

Solving one-step equations with multiplication and division Example 1: 3x = 12. Divide both sides by 3: x = 4. Example 2: x/4 = 5. Multiply both sides by 4: x = 20. 3x = 12 ÷ 3 both sides x = 4 x 4 = 5 × 4 both sides x = 20
3x = 12 is solved by dividing by 3; x/4 = 5 is solved by multiplying by 4.

If the equation is 3x = 12, x is being multiplied by 3. To isolate x, divide both sides by 3: 3x ÷ 3 = 12 ÷ 3, giving x = 4. Division is the inverse of multiplication, so it cancels the 3 on the left.

If the equation is x/4 = 5, x is being divided by 4. To isolate x, multiply both sides by 4: (x/4) × 4 = 5 × 4, giving x = 20. Multiplication is the inverse of division, so it cancels the 4 on the left.

Always substitute the solution back into the original equation to confirm it works: 3(4) = 12 √ and 20/4 = 5 √. This one-step process, along with modeling one-step equations, is the foundation for two-step linear equations and more complex equation types, which combine multiple inverse operations in sequence.

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