Multi-Step Equation
A multi-step equation requires three or more operations to solve, often involving the distributive property or combining like terms before isolating the variable.
Definition
A multi-step equation takes more than two steps to solve: you might need to use the distributive property, combine like terms, and then apply inverse operations, especially when variables appear on both sides. The standard strategy is to simplify each side, collect variables on one side, collect constants on the other, and then solve. Every multi-step linear equation in one variable reduces to the form $ax + b = 0$ after full simplification; the "multi-step" label really refers to the algebraic manipulations required to reach that standard form, and if $a \neq 0$ there is exactly one solution. In linear algebra, systems of multi-step equations are solved simultaneously via row reduction (Gaussian elimination), exploiting the linearity of the operations.
Example
Solve $2(x + 3) + x = 15$: distribute to get $2x + 6 + x = 15$, combine like terms to get $3x + 6 = 15$, subtract $6$ to get $3x = 9$, then divide by $3$ to get $x = 3$. Solve $5x - 3 = 2x + 9$: subtract $2x$ from both sides to get $3x - 3 = 9$, add $3$ to get $3x = 12$, then divide by $3$ to get $x = 4$. The equation $3(2x - 1) - 2(x + 4) = x + 5$ simplifies to $6x - 3 - 2x - 8 = x + 5$, then $4x - 11 = x + 5$, then $3x = 16$, so $x = 16/3$.
Key Insight
Multi-step equations combine all the skills you have learned, so work through them one step at a time and do not rush. When variables appear on both sides, it is a sign that the equation may describe a situation where two processes produce the same result at some value of $x$, and the multi-step process is really just algebraic preprocessing to reveal the canonical form $ax = b$.