Solving Two-Step Equations
Undo the operations in reverse order, turn a word problem into an equation, and rearrange a formula for the variable you actually want.
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Before diving into the practice questions, take a moment to think about what you already know about this topic.
Know
- Two-step equations require two inverse operations applied in reverse order
- The structure of an equation of the form $ax + b = c$
Understand
- Undo operations using SADMEP (the reverse of BEDMAS), undo $+/-$ first, then $\times/\div$
- Why the order of undoing matters
Can Do
- Solve equations of the form $ax + b = c$ in two clear steps
- Check answers by substituting back into the original equation
To solve $ax + b = c$, you undo the operations in reverse order the last operation applied is the first one you undo.
Think of it like getting dressed: you put your socks on before shoes, so you take your shoes off before socks. In $3x + 7$, the variable is first multiplied by 3, then 7 is added. To undo: first subtract 7, then divide by 3.
Watch each example. Every step is labelled so you know exactly why each move is made.
Example 1, $3x + 7 = 19$
| Working | Why? |
|---|---|
| $3x + 7 = 19$ | Start with the equation. |
| $3x + 7 - 7 = 19 - 7$ | Step 1: subtract 7 from both sides (undo the $+7$). |
| $3x = 12$ | Simplify. |
| $3x \div 3 = 12 \div 3$ | Step 2: divide both sides by 3 (undo the $\times 3$). |
| $x = 4$ | Solution found. |
| Check: $3(4) + 7 = 12 + 7 = 19$ ✓ | Substituting back confirms $x = 4$. |
Example 2, $2x - 5 = 11$
| Working | Why? |
|---|---|
| $2x - 5 = 11$ | Start with the equation. |
| $2x - 5 + 5 = 11 + 5$ | Step 1: add 5 to both sides (undo the $-5$). |
| $2x = 16$ | Simplify. |
| $2x \div 2 = 16 \div 2$ | Step 2: divide both sides by 2 (undo the $\times 2$). |
| $x = 8$ | Solution found. |
| Check: $2(8) - 5 = 16 - 5 = 11$ ✓ | Substituting back confirms $x = 8$. |
Real-world problems can be turned into two-step equations. The key is to identify the unknown, translate the words into algebra, then solve.
Problem: A taxi charges $2 per km plus a $5 flag fall. A trip costs $27 in total. How far did the taxi travel?
Let $d$ = distance in km. The equation is: $$2d + 5 = 27$$
Subtract 5: $2d = 22$. Divide by 2: $d = 11$ km.
A formula becomes an equation after you substitute the given values. Keep the unknown as a pronumeral, then solve the resulting equation.
- Substitute the known values
$$46=2l+2(8)$$
- Simplify to form a two-step equation
$$46=2l+16$$
- Solve for the remaining pronumeral
$$30=2l \quad \Rightarrow \quad l=15$$
- Verify in the original formula
$$2(15)+2(8)=30+16=46 \quad \checkmark$$
Watch Me Solve It · Worked example
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1Add 3 to both sides$5x - 3 + 3 = 22 + 3 \quad \Rightarrow \quad 5x = 25$The constant is $-3$. To undo subtraction, we add 3 to both sides. This isolates the $5x$ term (SADMEP: undo $+/-$ first).
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2Divide both sides by 5$5x \div 5 = 25 \div 5 \quad \Rightarrow \quad x = 5$The coefficient is 5. Dividing both sides by 5 undoes the multiplication and gives us $x$ on its own.
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3Check by substituting $x = 5$$5(5) - 3 = 25 - 3 = 22$ ✓Always check! Substitute your answer back into the original equation. Both sides equal 22, so $x = 5$ is correct.
Quick Check · 5 questions
Q1. Solve $7x - 9 = 26$. Show each step clearly and check your answer.
Q2. Solve $\frac{x}{3} + 8 = 5$. Show each step and check your answer.
Q3. A gym membership costs a $\$25$ joining fee plus $\$15$ per week. If Maria paid $175 in total for her first period, write and solve an equation to find how many weeks she attended.
Undo a fixed fee and a repeated cost
Undo a fixed fee and a repeated cost. A club charges a fixed booking fee of 7 dollars plus 5 dollars per participant. The total bill is 42 dollars.
Form and solve a two-step equation for the number of participants. A student divides 42 by 5 before subtracting 7. Explain why that order is wrong and check your solution in the original situation.
Show your working first, then open the worked solution to check your reasoning.
Check the worked solution
The equation is $5n+7=42$. Subtract the fixed fee first: $5n=35$. Then divide by 5: $n=7$ participants. Check: $5(7)+7=42$ dollars. The fee is added after the per-participant cost is calculated, so it must be undone first. Dividing the whole bill by 5 also divides the fixed fee, so subtracting 7 afterwards does not reverse the original operations.
Key Concept
Review the main ideas from this lesson.
Formulas
Key formulas and rules.
Watch Out
Common mistakes to avoid.
Check
Always verify your answers.
Practice
Keep practicing to master.
Next
Build on these skills.
Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
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Brain Trainer
Set a timer for 4 minutes. Solve as many as you can. Check your answers after.