Solving Multi-Step Equations

Multi-step equations combine brackets, like terms, fractions, and variables on both sides while preserving equality through reversible operations.

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Model → preserve balance → check. A modelling step leads through equivalent operations to an isolated unknown, then loops back through substitution. For inequalities, the method label is a reminder to justify any change in comparison direction separately.

Solving an equation means using reversible operations to produce an equivalent equation in which the variable is isolated.

Equality is the invariant

An equation is a balance. Adding, subtracting, multiplying, or dividing both sides by the same permitted value preserves the solution set. Each line should follow from the one before it, so the reasoning can be checked locally.

The goal is not “move it and change the sign.” That phrase hides the operation. Write what happens to both sides.

A reliable order

Simplify each side: clear brackets and combine like terms.
Clear fractions if doing so makes the equation easier.
Collect variable terms on one side.
Collect constants on the other side.
Divide by the variable’s coefficient.
Substitute into the original equation to check.

This is a flexible guide, not a law. An obvious division before distribution may save work.

Brackets and like terms

Check: $3(12-5)+4=3(7)+4=25$.

Variables on both sides

Solve $5x+7=2x+25$. Subtract $2x$ from both sides and subtract $7$ from both sides:

$$ 3x=18, $$

so $x=6$. Moving the smaller variable term often keeps the remaining coefficient positive, which can make arithmetic calmer.

Common mistakes

Changing one side only. Every balance-preserving operation must affect both sides.

Distributing incompletely. Multiply the outside factor by every bracketed term.

Clearing only some denominators. The common multiplier applies to each term in the equation.

Stopping when the variable has a coefficient. $4x=20$ is not isolated; divide to get $x=5$.

Checking only the final simplified line. Substitute into the original equation, where transcription errors can be detected.

Quick self-check

Is each side simplified before I collect terms?
Did each operation affect both sides?
Could the equation have all real solutions or none?
Does substitution make the original left and right sides equal?