Two-Step Equations

Solving Two-Step Equations

A two-step equation has the form ax + b = c, where you need exactly two operations to isolate the variable. The strategy is always the same: undo addition/subtraction first, then undo multiplication/division — the reverse of the order of operations.

The Golden Rule of Equations

Whatever you do to one side of an equation, you must do to the other side too. This keeps the two sides equal, just like a balanced scale — if you add weight to one side, you must add the same weight to the other side to keep it level.

The Two Steps

1. Undo addition or subtraction: add or subtract the constant term from both sides so the variable term is alone on one side.
2. Undo multiplication or division: divide (or multiply) both sides by the coefficient of the variable to get the variable completely alone.

Example

Solve 3x + 7 = 22 for x.

1. Subtract 7 from both sides to undo the addition: 3x + 7 - 7 = 22 - 7 \;\Rightarrow\; 3x = 15
2. Divide both sides by 3 to undo the multiplication: \frac{3x}{3} = \frac{15}{3} \;\Rightarrow\; x = 5

Checking Your Answer

Always substitute your solution back into the original equation to verify it. With x = 5: 3(5) + 7 = 15 + 7 = 22, which matches — so x = 5 is correct.

Equations with Negative Coefficients or Division

The same two steps work even when the coefficient is negative or when the variable is divided by a number, such as \dfrac{x}{4} - 3 = 5. Undo the -3 first by adding 3 to both sides (\dfrac{x}{4} = 8), then undo the division by multiplying both sides by 4 (x = 32).

A Second Example

Solve -2x - 5 = 9.

1. Add 5 to both sides: -2x = 14
2. Divide both sides by -2: x = -7

Check: -2(-7) - 5 = 14 - 5 = 9, which matches the original equation, so x = -7 is correct. Notice that dividing by a negative number here did not require flipping any sign, since we are solving an equation, not an inequality — that special flipping rule only applies to inequalities, which you'll see in a later lesson.

Key Takeaways

• A two-step equation needs one addition/subtraction step and one multiplication/division step.
• Undo addition/subtraction first, then undo multiplication/division — the reverse of PEMDAS.
• Whatever operation you perform, apply it to both sides of the equation.
• Always check your solution by substituting it back into the original equation.