Two-Step Equations

Now that you can solve one-step equations, it is time to handle equations that require two steps to isolate the variable. These equations have two operations applied to the variable — typically a multiplication (or division) combined with an addition (or subtraction). The strategy is to undo the operations in reverse order: deal with addition or subtraction first, then multiplication or division.

The Two-Step Strategy

For an equation like $ax + b = c$:

1. Undo addition or subtraction — move the constant term to the other side
2. Undo multiplication or division — isolate the variable completely

This is the reverse of the order of operations. When the expression $2x + 3$ was built, the multiplication happened first ($2 \times x$), then the addition ($+3$). To undo it, reverse the order: subtract 3, then divide by 2.

Worked Examples

Example 1: Solve $2x + 5 = 13$

Step 1 — Subtract 5 from both sides:

$2x + 5 - 5 = 13 - 5$

$2x = 8$

Step 2 — Divide both sides by 2:

$\frac{2x}{2} = \frac{8}{2}$

$x = 4$

Check: $2(4) + 5 = 8 + 5 = 13$ — that checks out.

Example 2: Solve $3n - 7 = 20$

Step 1 — Add 7 to both sides:

$3n - 7 + 7 = 20 + 7$

$3n = 27$

Step 2 — Divide both sides by 3:

$n = \frac{27}{3} = 9$

Check: $3(9) - 7 = 27 - 7 = 20$ — correct.

Example 3: Solve $\frac{x}{4} + 3 = 10$

Step 1 — Subtract 3 from both sides:

$\frac{x}{4} = 7$

Step 2 — Multiply both sides by 4:

$x = 28$

Check: $\frac{28}{4} + 3 = 7 + 3 = 10$ — confirmed.

Example 4: Solve $-5y + 12 = -8$

Step 1 — Subtract 12 from both sides:

$-5y = -8 - 12$

$-5y = -20$

Step 2 — Divide both sides by $-5$:

$y = \frac{-20}{-5} = 4$

Check: $-5(4) + 12 = -20 + 12 = -8$ — correct.

Example 5: Solve $\frac{m}{3} - 8 = 2$

Step 1 — Add 8 to both sides:

$\frac{m}{3} = 10$

Step 2 — Multiply both sides by 3:

$m = 30$

Check: $\frac{30}{3} - 8 = 10 - 8 = 2$ — confirmed.

When the Variable Is on the Right

The variable does not always have to be on the left. The same steps apply.

Example 6: Solve $17 = 4a + 1$

Step 1 — Subtract 1 from both sides:

$16 = 4a$

Step 2 — Divide both sides by 4:

$4 = a$

So $a = 4$.

Check: $4(4) + 1 = 17$ — correct.

Equations with Negative Coefficients

Example 7: Solve $-2x + 9 = 1$

Step 1 — Subtract 9:

$-2x = 1 - 9 = -8$

Step 2 — Divide by $-2$:

$x = \frac{-8}{-2} = 4$

Check: $-2(4) + 9 = -8 + 9 = 1$ — confirmed.

Equations with Decimals

Example 8: Solve $1.5x + 4 = 13$

Step 1 — Subtract 4:

$1.5x = 9$

Step 2 — Divide by 1.5:

$x = \frac{9}{1.5} = 6$

Check: $1.5(6) + 4 = 9 + 4 = 13$ — correct.

Word Problems Leading to Two-Step Equations

Many real-life situations naturally produce two-step equations. The key is to translate the words into an equation, then solve.

Example 9: Plumber’s Bill

A plumber charges a $45 service fee plus $30 per hour. The total bill was $165. How many hours did the plumber work?

Let $h$ represent the hours worked:

$30h + 45 = 165$

Step 1 — Subtract 45:

$30h = 120$

Step 2 — Divide by 30:

$h = 4$

Answer: The plumber worked 4 hours.

Check: $30(4) + 45 = 120 + 45 = 165$ — confirmed.

Real-World Application: Electrician — Wire Length Calculation

An electrician needs to run wire from a junction box to several outlets. The job requires a 15-foot lead from the box, plus 8 feet of wire for each outlet. If the electrician has 63 feet of wire total, how many outlets can be wired?

Let $n$ represent the number of outlets:

$8n + 15 = 63$

Step 1 — Subtract 15:

$8n = 48$

Step 2 — Divide by 8:

$n = 6$

Answer: The electrician can wire 6 outlets with 63 feet of wire.

Check: $8(6) + 15 = 48 + 15 = 63$ — correct.

Real-World Application: Nursing — IV Infusion Time

A nurse starts an IV with 50 mL already infused. The pump delivers 125 mL per hour. The total volume to be infused is 800 mL. How many more hours will the infusion take?

Let $t$ represent the remaining hours:

$125t + 50 = 800$

Step 1 — Subtract 50:

$125t = 750$

Step 2 — Divide by 125:

$t = 6$

Answer: The infusion will take 6 more hours.

Check: $125(6) + 50 = 750 + 50 = 800$ — confirmed.

Common Mistakes to Avoid

1. Dividing before subtracting. In $2x + 5 = 13$, do NOT divide everything by 2 first. While technically possible (you would need to divide ALL three terms), it creates fractions and is error-prone. Subtract the constant first — it is cleaner.

2. Subtracting from only one side. The balance must be maintained. If you subtract 5 from the left, subtract 5 from the right too.

3. Sign errors when subtracting a negative. In $3x - 8 = 10$, add 8 to both sides: $3x = 18$. A common error is writing $3x = 2$.

4. Forgetting to check. Two steps means two chances for arithmetic errors. Always plug your answer back into the original equation.

5. Misidentifying the operations. Before solving, identify what is being done to the variable. In $\frac{x}{4} + 3 = 10$, the variable is divided by 4 and then 3 is added. Reverse: subtract 3, then multiply by 4.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Two-step equations require two inverse operations to isolate the variable
• Always undo addition or subtraction first, then multiplication or division — this is the reverse of PEMDAS
• Whatever you do to one side, you must do to the other to keep the equation balanced
• Always check your answer by substituting it back into the original equation
• Word problems in trades and healthcare frequently produce two-step equations — recognizing the pattern is the first step to solving them
• Two-step equations are the gateway to multi-step equations in Algebra 1

Return to Pre-Algebra for more topics in this section.