Two-Step Equations

Two-Step Equations

A two-step equation — such as 3x + 5 = 20 — requires undoing two operations to isolate the variable. The key skill is figuring out the correct order in which to undo them.

Undo Operations in Reverse Order

Think about how the expression 3x + 5 was built, step by step, starting from x: first x is multiplied by 3, and then 5 is added. To solve the equation, undo those steps in the opposite order they were applied — last operation first:

1. Undo the addition/subtraction first (it was applied last).
2. Undo the multiplication/division second (it was applied first).

This mirrors "reverse PEMDAS": building an expression follows Parentheses → Exponents → Multiply/Divide → Add/Subtract, so unbuilding it goes in reverse: Add/Subtract → Multiply/Divide → Exponents → Parentheses.

Worked Pattern

To solve ax + b = c:

1. Subtract b from both sides: ax = c - b.
2. Divide both sides by a: x = \frac{c-b}{a}.

For example, solve 3x + 5 = 20: subtract 5 from both sides to get 3x = 15, then divide by 3 to get x = 5.

Equations With Subtraction or Division First

The same reverse-order idea applies no matter which operations appear. For \frac{x}{4} - 2 = 6: first undo the subtraction by adding 2 to both sides, giving \frac{x}{4} = 8; then undo the division by multiplying both sides by 4, giving x = 32.

Combining Like Terms First

Sometimes an equation isn't in the clean form ax+b=c right away, and you must simplify one or both sides first. For 2x + 3x - 4 = 21, first combine 2x + 3x = 5x to get 5x - 4 = 21, and then solve as a normal two-step equation: 5x = 25, so x = 5.

Example

Solve for x: -2x + 9 = -7.

Step 1: Undo the addition of 9 by subtracting 9 from both sides: -2x = -7 - 9 = -16

Step 2: Undo the multiplication by -2 by dividing both sides by -2: x = \frac{-16}{-2} = 8

Step 3 (check): -2(8) + 9 = -16 + 9 = -7, which matches the original right side. The solution x = 8 is correct.

Key Takeaways

• A two-step equation requires undoing two operations, in the reverse of the order they were applied.
• Undo addition/subtraction first, then undo multiplication/division.
• Combine like terms on each side before treating the equation as a standard two-step form.
• Always check your final answer by substituting it back into the original equation.