Two-Step Equations with Integers

Same two moves as before — now the numbers can be negative.

▶ Practice this concept

🧱 See It

Level 2's balance scale still works exactly the same way: undo addition/subtraction first, then undo multiplication/division. The only thing that's new is that some of the numbers involved can be negative — the loose blocks, the total on the other side, even the answer itself. Equations like 3x − 7 = −13 or −2x + 5 = 11 look scarier, but they balance and unbalance the exact same way.

x
→
−3

A mystery bag can still equal a negative number when you finally solve for it. Nothing about the scale changes — only the sign bookkeeping is new.

✏️ Draw It

Solve −2x + 5 = 11. Two bags, each worth −x, plus 5 loose blocks sit on the left. 11 loose blocks sit on the right:

−x
−x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

Move 1: Remove 5 blocks from both sides — exactly like before.

−x
−x
1
1
1
1
1
1

Now 2 bags of −x balance 6 blocks. Because the coefficient is negative (−2x), the last move — dividing both sides by −2 — needs a sign check that a positive coefficient wouldn't. That move is worked out fully in the Write It section below.

🔢 Write It

−2x + 5 = 11

Move 1 — undo the addition (subtract 5 from both sides):

−2x = 6

Move 2 — undo the multiplication (divide both sides by −2):

x = −3

Dividing by a negative number is the exact same mechanical division as always — 6 ÷ 2 = 3. Then apply the same-sign/different-sign rule from Level 2, separately, to fix the sign: positive ÷ negative is different signs, so the answer is negative.

Divide the numbers first, ignoring signs. THEN apply the same-sign/different-sign rule as its own separate step — never blend the sign and the division into one.

💡 Worked Examples

Example 1: 3x − 7 = −13

  1. Add 7 to both sides: 3x = −6
  2. Check the signs for the division: −6 ÷ 3 is different signs, so the answer will be negative.
  3. Divide the numbers: 6 ÷ 3 = 2, so x = −2.

Example 2: −4x + 9 = 1

  1. Subtract 9 from both sides: −4x = −8
  2. Check the signs for the division: −8 ÷ (−4) is the same sign, so the answer will be positive.
  3. Divide the numbers: 8 ÷ 4 = 2, so x = 2.

🧠 Quick Reference

Same two moves as Level 2: undo addition/subtraction first, then undo multiplication/division. The only new step is applying the same-sign/different-sign division rule at the very last step — do the sign and the division as two separate checks, never blended into one.
👪 For Parents & Tutors

BC Curriculum (Level 3 / Grade 8, Algebra & Patterning): "Solve two-step equations with integer coefficients, constants, and solutions."

Watch for: forgetting to apply the integer sign rules (same-sign/different-sign) at the final division step is the main trap here, especially when the coefficient itself is negative (e.g. dividing by −4). Treat the sign-check as its own explicit step, separate from "do the division" — exactly the pattern already established on Level 2's integer pages. Doing the sign and the arithmetic in the same mental step is where it goes wrong under working-memory load.

Comes up again: this depends directly on Level 2's integer operations and two-step equations, and sets up Level 4 (Grade 9) multi-step equations with variables on both sides.