Multi-Step Equation
Pre-AlgebraA 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 do inverse operations.
Example
Solve $2(x + 3) + x = 15$. Distribute: $2x + 6 + x = 15$. Combine like terms: $3x + 6 = 15$. Subtract $6$: $3x = 9$. Divide by $3$: $x = 3$.
Key Insight
Multi-step equations combine all the skills you have learned. Work through them one step at a time and do not rush.
Definition
A multi-step equation requires combining like terms, applying the distributive property, or both, before isolating the variable. Variables may appear on both sides of the equation. The standard strategy is: simplify each side, collect variables on one side, collect constants on the other, then solve.
Example
Solve $5x - 3 = 2x + 9$. Subtract $2x$ from both sides: $3x - 3 = 9$. Add $3$ to both sides: $3x = 12$. Divide by $3$: $x = 4$.
Key Insight
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 $x$.
Definition
Multi-step linear equations are still of the form $ax + b = 0$ after full simplification; the "multi-step" label refers to the algebraic manipulations required to reach that standard form. In linear algebra, systems of multi-step equations are solved simultaneously via row reduction (Gaussian elimination), exploiting the linearity of the operations.
Example
The equation $3(2x - 1) - 2(x + 4) = x + 5$ simplifies as: $6x - 3 - 2x - 8 = x + 5$, giving $4x - 11 = x + 5$, then $3x = 16$, so $x = 16/3$.
Key Insight
Every multi-step linear equation in one variable reduces to the form $ax = b$. If $a \neq 0$, there is exactly one solution. The multi-step process is just algebraic preprocessing to reveal this canonical form.