Two-Step Equation
Pre-AlgebraA two-step equation requires exactly two inverse operations to isolate the variable and find the solution.
Definition
A two-step equation needs two moves to solve. First you add or subtract, then you multiply or divide (or the other way around) until the variable is alone.
Example
Solve $2x + 3 = 11$. Step 1: subtract $3$ from both sides: $2x = 8$. Step 2: divide both sides by $2$: $x = 4$.
Key Insight
Think of the variable as being wrapped in two layers. You peel the outer layer off first, then the inner layer.
Definition
A two-step equation is a linear equation that requires two inverse operations to isolate the variable. The standard form is $ax + b = c$. The procedure is: (1) undo addition or subtraction, then (2) undo multiplication or division.
Example
Solve $x/3 - 5 = 2$. Step 1: add $5$ to both sides: $x/3 = 7$. Step 2: multiply both sides by $3$: $x = 21$. Check: $21/3 - 5 = 7 - 5 = 2$. Correct.
Key Insight
The two steps always follow the reverse of the order of operations: undo addition/subtraction first (outer operations), then undo multiplication/division (inner operations).
Definition
A two-step linear equation $ax + b = c$ ($a \neq 0$) has the unique solution $x = (c-b)/a$ over any field. This is derived by applying the additive inverse of $b$ and then the multiplicative inverse of $a$. The existence and uniqueness of the solution reflect the field axioms.
Example
Over $\mathbb{Q}$, solving $(3/4)x + 1/2 = 5/4$: subtract $1/2$: $(3/4)x = 3/4$. Multiply by $4/3$: $x = 1$. The solution is unique in any field of characteristic $0$.
Key Insight
In a field of characteristic $p$ (a prime), the equation $ax = b$ has no solution when $a = 0$ and $b \neq 0$, and every element as a solution when $a = 0$ and $b = 0$. This generalizes the "no solution / infinite solutions" cases of the real setting.