Two-Step Equation
In short: Two-step equation refers to an equation where the unknown has been multiplied and then shifted, so you free it by undoing an addition or subtraction first and a multiplication or division second.
7th GradeAlgebra7.EE.B.4a
What is Two-Step Equation?
A two-step equation has the shape \(px + q = r\): the unknown is first multiplied by \(p\), then shifted by \(q\). It models any situation that mixes a flat amount with a repeating amount, such as an entry fee plus a monthly charge.
Solving means walking backward through those operations. Strip the shift from both sides, then split both sides by the multiplier. Every move keeps the sides balanced, so the value you reach for \(x\) makes the original statement true.
How to spot it
- A term like 18m or x/4 sits beside a loose number that is added or subtracted.
- A story pairs a one-time fee with a per-unit rate and reveals the grand total.
- Undoing a single operation falls short because the unknown carries a multiplier and a companion constant.
- You are hunting for the missing input that produces a stated total.
The method
- Spot the constant attached to the variable term and cancel it with the opposite operation on each side.
- With the variable term alone, split each side by its coefficient.
- Write the solution as a statement such as \(m = 8\).
- Substitute the value into the original equation and confirm both sides match.
Two-Step Equation: solved example
Problem
A climbing gym asks \(\$25\) to sign up and \(\$18\) for every month after. Dara has handed over \(\$169\) altogether. How many months has she belonged?
Solution
Let \(m\) be the number of months: \(18m + 25 = 169\).
Subtract 25: \(18m = 144\). Divide by 18: \(m = 8\). Check: \(18 \times 8 + 25 = 169\). Dara has belonged for 8 months.
Try it
Question 1
Solve \(\frac{x}{4} – 6 = 3\).
Show the answer
Add 6: \(\frac{x}{4} = 9\). Multiply by 4: \(x = 36\).
Question 2
Solve \(-2y + 9 = -15\).
Show the answer
Subtract 9: \(-2y = -24\). Divide by \(-2\): \(y = 12\).
Two-Step Equation vs. One-Step Equation
A one-step equation such as \(x + 7 = 20\) needs a single inverse operation. A two-step equation has the variable wearing both a coefficient and a constant, so two moves are required. Order matters: clear the constant before dividing, or else divide every term, not only the variable term.
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