Mathematics • Year 10 • Unit 2 • Lesson 14

Simultaneous Equations (Elimination) — Skill Drill

Build fluency with elimination from Lesson 14: same signs → subtract; opposite signs → add; otherwise multiply to match coefficients first. Then back-substitute and check.

Build · I Do / We Do / You Do

1. I do — fully worked example

Same-sign coefficients eliminated by subtraction. Read each reason.

Problem. Solve simultaneously: 5x + 3y = 23 and 5x − 2y = 8.

Step 1 — Compare coefficients of the same variable.

x-coefficients: +5 and +5. Same sign → subtract.

Step 2 — Subtract Eq 2 from Eq 1 term-by-term.

(5x + 3y) − (5x − 2y) = 23 − 8

0 + 5y = 15 → y = 3

Reason: subtracting "− 2y" flips the sign to + 2y. So 3y − (−2y) = 3y + 2y = 5y.

Step 3 — Back-substitute y = 3 into either original.

5x + 3(3) = 23 → 5x = 14 → x = 14/5 = 2.8

Step 4 — Check in the OTHER original.

5(2.8) − 2(3) = 14 − 6 = 8 ✓

Answer: x = 2.8, y = 3.

Stuck? Revisit lesson § "Add or Subtract?" — Worked Example 1.

2. We do — fill in the missing steps

Opposite-sign coefficients eliminated by addition. 5 marks

Problem. Solve: 3x + 4y = 18 and 2x − 4y = 2.

Step 1 — Compare coefficients of y: +4 and −4. They are opposite signs, so we _____ the equations.

Step 2 — Add the equations term-by-term:

(3x + 4y) + (2x − 4y) = 18 + 2

____ x + ____ y = ____

Step 3 — Solve for x: ____ x = ____ → x = ____.

Step 4 — Back-substitute into Eq 1: 3( ____ ) + 4y = 18 → 4y = ____ → y = ____.

Step 5 — Check in Eq 2: 2( ____ ) − 4( ____ ) = ____ ✓.

Stuck? Revisit lesson § "Add or Subtract?" — Worked Example 2.

3. You do — independent practice

Decide whether to add, subtract, or multiply first. Show your reasoning at each step.

Foundation — direct add/subtract

3.1 x + y = 7 and x − y = 3. 1 mark

3.2 2x + y = 11 and 2x − y = 5. 1 mark

3.3 4x + 3y = 19 and 4x − 2y = 4. 1 mark

3.4 3x + 2y = 12 and 3x + 5y = 21. 1 mark

Standard — multiply one equation first

3.5 x + 2y = 10 and 3x − y = 9. (Multiply Eq 1 by 3 to match x.) 2 marks

3.6 2x + 3y = 14 and 3x + 5y = 22. (Multiply Eq 1 by 3 and Eq 2 by 2 to match x at LCM = 6.) 2 marks

Extension — multiply both equations

3.7 3x + 4y = 11 and 2x − 5y = 8. (Eliminate x: multiply Eq 1 by 2 and Eq 2 by 3.) 3 marks

3.8 5x − 2y = 1 and 3x + 4y = 11. (Multiply Eq 1 by 2 so the y-coefficients become opposites.) 3 marks

Stuck on 3.8? After × 2: 10x − 4y = 2 and 3x + 4y = 11. Add (opposite signs on y) → 13x = 13 → x = 1.

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Answers — Do not peek before attempting

Section 2 — We do (3x + 4y = 18; 2x − 4y = 2)

Step 1: add. Step 2: 5x + 0y = 20. Step 3: 5x = 20 → x = 4. Step 4: 3(4) + 4y = 18 → 4y = 6 → y = 1.5. Step 5: 2(4) − 4(1.5) = 8 − 6 = 2 ✓.

3.1 — x + y = 7; x − y = 3

Opposite sign on y, add: 2x = 10 → x = 5, y = 2. (5, 2).

3.2 — 2x + y = 11; 2x − y = 5

Add: 4x = 16 → x = 4, y = 3. (4, 3).

3.3 — 4x + 3y = 19; 4x − 2y = 4

Same sign on x, subtract Eq 2 from Eq 1: 5y = 15 → y = 3, x = 2.5. (2.5, 3).

3.4 — 3x + 2y = 12; 3x + 5y = 21

Same sign on x, subtract Eq 1 from Eq 2: 3y = 9 → y = 3, x = 2. (2, 3).

3.5 — x + 2y = 10; 3x − y = 9

× Eq 1 by 3: 3x + 6y = 30. Subtract Eq 2: 7y = 21 → y = 3, x = 10 − 6 = 4. (4, 3).

3.6 — 2x + 3y = 14; 3x + 5y = 22

× Eq 1 by 3: 6x + 9y = 42. × Eq 2 by 2: 6x + 10y = 44. Subtract: y = 2, then 2x = 8 → x = 4. (4, 2). Check Eq 2: 3(4) + 5(2) = 22 ✓.

3.7 — 3x + 4y = 11; 2x − 5y = 8

× Eq 1 by 2: 6x + 8y = 22. × Eq 2 by 3: 6x − 15y = 24. Subtract: 23y = −2 → y = −2/23. Back: 3x + 4(−2/23) = 11 → 3x = 11 + 8/23 = 261/23 → x = 87/23 ≈ 3.78. Check Eq 2: 2(87/23) − 5(−2/23) = 174/23 + 10/23 = 184/23 = 8 ✓.

3.8 — 5x − 2y = 1; 3x + 4y = 11

× Eq 1 by 2: 10x − 4y = 2. Add to Eq 2: 13x = 13 → x = 1. Back: 3(1) + 4y = 11 → y = 2. (1, 2). Check Eq 1: 5(1) − 2(2) = 1 ✓.