Key takeaway: A two-step equation is an equation that takes exactly two inverse operations, one to cancel a constant and one to cancel a coefficient, before the variable stands alone.
Grade 77.EE.B.4aExpressions and Equations
Definition of Two-Step Equation
A two-step equation hides its unknown behind two operations. The unknown was multiplied by a number, and then another number was added or taken away, as in \(2.25m + 3.50 = 17\).
Peel the layers off in reverse order, like taking off shoes before socks because socks went on first. The last operation applied is the first one you remove. A constant term goes before the coefficient. Keep the scale balanced by treating both sides the same way each time.
Formula and notation
The standard shape of a two-step equation is
\[ax + b = c\]
with \(a \neq 0\). Subtract \(b\) from both sides, then divide both sides by \(a\), giving \(x = \frac{c – b}{a}\).
Key facts
- Cancel the constant term first; the coefficient comes second.
- Whatever you do on the left of the equal sign, repeat on the right.
- Plug the solution back into the original equation as a final check.
- A flat fee plus a price per item is the classic story behind this form.
Two-Step Equation example
A cab in Austin starts the meter at \$3.50 and adds \$2.25 per mile. Nadia paid exactly \$17.00. How many miles was her ride? Solve \(2.25m + 3.50 = 17\).
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Subtract 3.50 from both sides: \(2.25m = 13.50\).
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Divide both sides by 2.25: \(m = \frac{13.50}{2.25} = 6\).
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Check: \(2.25 \times 6 + 3.50 = 13.50 + 3.50 = 17.00\). It works.
Nadia rode 6 miles.
Common mistake
Watch out
To solve \(4x – 13 = 27\), Theo divides by 4 first and gets \(x – 13 = 6.75\), then \(x = 19.75\).
Why it is wrong
Theo divided \(4x\) and 27 by 4 but forgot the 13, so the equation is no longer balanced. Either divide every term by 4, or undo the subtraction first: \(4x = 40\), so \(x = 10\). Check: \(4 \times 10 – 13 = 27\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Expressions and Equations