Solving Two-Step Equations Step by Step

Grade: 7-8 | Topic: Algebra

What You Will Learn

After this lesson you will know how to recognize a two-step equation, understand the "undo" strategy for choosing which operation to apply first, and solve equations of the form $ax + b = c$ with confidence. You will also practice checking your answers and avoiding the most common sign errors.

Theory

What is a two-step equation?

A two-step equation is an equation that requires exactly two inverse operations to isolate the variable. The standard form is:

$ax + b = c$

where $a$ is the coefficient (multiplier) of $x$, $b$ is a constant added or subtracted, and $c$ is the result on the other side.

Examples of two-step equations:

• $3x + 7 = 22$
• $\dfrac{x}{4} - 5 = 3$
• $-2x + 9 = 1$

The "undo" strategy — reverse order of operations

When an equation was "built," the operations were applied in a certain order. To solve it, you reverse that order:

1. First, undo addition or subtraction (the outermost operation).
2. Then, undo multiplication or division (the operation closest to the variable).

Think of putting on socks and shoes: you put socks on first, then shoes. To take them off, you remove shoes first, then socks. Equations work the same way.

For $ax + b = c$:

$\text{Step 1: } ax + b - b = c - b \implies ax = c - b$

$\text{Step 2: } \frac{ax}{a} = \frac{c - b}{a} \implies x = \frac{c - b}{a}$

Why this order matters

If you tried to divide first in an equation like $3x + 7 = 22$, you would need to divide every term by 3:

$x + \frac{7}{3} = \frac{22}{3}$

This creates fractions that make the problem harder. By subtracting 7 first, you keep the numbers clean:

$3x = 15 \implies x = 5$

Always undo the addition/subtraction first to keep things simple.

Worked Examples

Example 1: Standard form — positive coefficient (easy)

Problem: Solve $2x + 5 = 17$

Step 1: Subtract 5 from both sides to undo the addition.

$2x + 5 - 5 = 17 - 5$

$2x = 12$

Step 2: Divide both sides by 2 to undo the multiplication.

$\frac{2x}{2} = \frac{12}{2}$

$x = 6$

Answer: $x = 6$

Check: $2(6) + 5 = 12 + 5 = 17$ ✓

Example 2: Subtraction in the equation (easy)

Problem: Solve $4x - 9 = 19$

Step 1: Add 9 to both sides to undo the subtraction.

$4x - 9 + 9 = 19 + 9$

$4x = 28$

Step 2: Divide both sides by 4.

$\frac{4x}{4} = \frac{28}{4}$

$x = 7$

Answer: $x = 7$

Check: $4(7) - 9 = 28 - 9 = 19$ ✓

Example 3: Division in the equation (medium)

Problem: Solve $\dfrac{x}{3} + 8 = 14$

Step 1: Subtract 8 from both sides.

$\frac{x}{3} + 8 - 8 = 14 - 8$

$\frac{x}{3} = 6$

Step 2: Multiply both sides by 3 to undo the division.

$\frac{x}{3} \times 3 = 6 \times 3$

$x = 18$

Answer: $x = 18$

Check: $\frac{18}{3} + 8 = 6 + 8 = 14$ ✓

Example 4: Negative coefficient (medium)

Problem: Solve $-5x + 12 = -8$

Step 1: Subtract 12 from both sides.

$-5x + 12 - 12 = -8 - 12$

$-5x = -20$

Step 2: Divide both sides by $-5$.

$\frac{-5x}{-5} = \frac{-20}{-5}$

$x = 4$

Answer: $x = 4$

Check: $-5(4) + 12 = -20 + 12 = -8$ ✓

Example 5: Negative result (challenging)

Problem: Solve $7x + 30 = 2$

Step 1: Subtract 30 from both sides.

$7x + 30 - 30 = 2 - 30$

$7x = -28$

Step 2: Divide both sides by 7.

$\frac{7x}{7} = \frac{-28}{7}$

$x = -4$

Answer: $x = -4$

Check: $7(-4) + 30 = -28 + 30 = 2$ ✓

Common Mistakes

Mistake 1: Dividing before subtracting

❌ $3x + 6 = 18 \implies x + 6 = 6 \implies x = 0$ (divided only the $3x$ term by 3, not the 6)

✅ $3x + 6 = 18 \implies 3x = 12 \implies x = 4$

Why this matters: If you divide first, you must divide every term on both sides. It is easier and safer to undo the addition or subtraction first.

Mistake 2: Sign errors when subtracting a negative

❌ $2x - 7 = 11 \implies 2x = 11 - 7 = 4$ (kept the minus sign instead of adding)

✅ $2x - 7 = 11 \implies 2x = 11 + 7 = 18 \implies x = 9$

Why this matters: To undo "$- 7$" you add 7 to both sides. The sign changes when a term moves across the equals sign. This is the number-one source of errors in equation solving.

Mistake 3: Forgetting to divide the negative sign

❌ $-4x = 20 \implies x = 20 \div 4 = 5$ (ignored the negative)

✅ $-4x = 20 \implies x = 20 \div (-4) = -5$

Why this matters: The coefficient includes its sign. When you divide by $-4$, the result is negative. A positive divided by a negative gives a negative answer.

Practice Problems

Try these on your own before checking the answers:

1. $5x + 3 = 28$
2. $3x - 11 = 7$
3. $\dfrac{x}{6} + 4 = 10$
4. $-2x + 15 = 3$
5. $8x - 20 = -4$

Click to see answers

1. $x = 5$ — Subtract 3: $5x = 25$, divide by 5: $x = 5$.
2. $x = 6$ — Add 11: $3x = 18$, divide by 3: $x = 6$.
3. $x = 36$ — Subtract 4: $\frac{x}{6} = 6$, multiply by 6: $x = 36$.
4. $x = 6$ — Subtract 15: $-2x = -12$, divide by $-2$: $x = 6$.
5. $x = 2$ — Add 20: $8x = 16$, divide by 8: $x = 2$.

Summary

• A two-step equation has the form $ax + b = c$ and needs two inverse operations to solve.
• Always undo addition/subtraction first, then undo multiplication/division (reverse order of operations).
• Pay careful attention to signs — especially when subtracting negatives or dividing by negative coefficients.
• Check your answer by substituting it back into the original equation.

Related Topics

• Linear Equations — How to Solve Step by Step
• Solving One-Step Equations — Examples and Practice
• How to Solve Equations with Fractions

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