Solving Multi-Step Equations

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

1. Simplify each side: clear brackets and combine like terms.
2. Clear fractions if doing so makes the equation easier.
3. Collect variable terms on one side.
4. Collect constants on the other side.
5. Divide by the variable’s coefficient.
6. 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:

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

Clearing fractions

Multiply every term on both sides by the least common denominator. For

the least common denominator is $6$. Multiplying the entire equation by $6$ gives $2x+3=5$, so $2x=2$ and $x=1$.

The multiplier applies to every term, not only the fractions that look inconvenient.

Identities and contradictions

Sometimes the variable disappears. If simplification produces a true statement such as $8=8$, every real value satisfies the original equation. If it produces a false statement such as $8=11$, no value satisfies it.

For example, $2(x+3)=2x+6$ is an identity, while $2(x+3)=2x+9$ is a contradiction.

Equations from context

Define the variable with units before writing the equation. If a gym charges a $25$ fee plus $12$ dollars per month and the total is $109$, let $m$ be the number of months:

Then $12m=84$, so $m=7$ months. The unit and context rule out interpretations that do not make sense.

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?

Related topics

• Distributive Property
• Combining Like Terms
• Linear Equation
• Literal Equations