Mathematics • Year 8 • Unit 2 • Lesson 19
Simultaneous Equations, Elimination Method
Build fluency with the sign rule: opposite signs ADD, same signs SUBTRACT. One worked example with direct elimination, one guided example where you have to subtract, then eight independent problems graduated from direct add/subtract to "multiply one equation first".
1. I do, fully worked example
Read every line. The sign rule: opposite signs ADD, same signs SUBTRACT. The aim is to make one variable cancel to zero.
Problem. Solve x + y = 7 and x − y = 3 by elimination.
Step 1, Line the equations up and inspect the y-terms.
Eq 1: x + y = 7
Eq 2: x − y = 3
Reason: y has coefficients +1 and −1, OPPOSITE signs. Add the equations to cancel y.
Step 2, Add the two equations (LHS + LHS = RHS + RHS).
(x + y) + (x − y) = 7 + 3
2x = 10
Reason: +y and −y combine to 0, leaving an equation with one variable.
Step 3, Solve for x.
x = 5
Step 4, Back-substitute into either original to find y.
Eq 1: 5 + y = 7 → y = 2
Step 5, VERIFY in the OTHER equation.
Eq 2: 5 − 2 = 3 ✓
Answer: (x, y) = (5, 2).
2. We do, fill in the missing steps
This one needs SUBTRACTION (same-sign coefficients). Fill every blank. 5 marks
Problem. Solve 2x + 3y = 13 and 2x + y = 7 by elimination.
Step 1, Inspect the x-terms: both are +2x. Same signs → ______________ the equations.
Step 2, Subtract Eq 2 from Eq 1 (carefully, distribute the minus):
(2x + 3y) − (2x + y) = 13 − ______
2x − 2x + 3y − y = ______
______y = ______
Step 3, Solve for y:
y = ______
Step 4, Back-substitute into Eq 2 to find x:
2x + ______ = 7 → 2x = ______ → x = ______
Step 5, Verify in Eq 1: 2(______) + 3(______) = ______ ✓ ?
Answer: (x, y) = (______, ______).
3. You do, independent practice
Show every step. 3.1–3.3 are foundation (direct add or subtract). 3.4–3.6 are standard (read the sign rule, then operate). 3.7–3.8 are extension (multiply one equation first).
Foundation, direct add or subtract
3.1 x + y = 9 and x − y = 3. Add to eliminate y. 2 marks
3.2 2x + y = 11 and 2x − y = 5. Add to eliminate y. 2 marks
3.3 3x + 2y = 16 and 3x − y = 10. Subtract to eliminate x. 2 marks
Standard, choose add or subtract
3.4 x + 2y = 8 and x + y = 5. (Same +x sign, subtract.) 3 marks
3.5 4x − 3y = 6 and 2x + 3y = 12. (Opposite y signs, add.) 3 marks
3.6 5x + 3y = 21 and 2x + 3y = 12. (Same +3y, subtract.) 3 marks
Extension, multiply one equation first
3.7 3x + 2y = 12 and x − y = 1. Multiply Eq 2 by 2 so the y-coefficients become +2y and −2y, then add. 3 marks
3.8 x + 4y = 14 and 3x − 2y = 0. Multiply Eq 2 by 2 so the y-coefficients become +4y and −4y, then add. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do 2x + 3y = 13 and 2x + y = 7
Step 1: SUBTRACT.
Step 2: (2x + 3y) − (2x + y) = 13 − 7 → 2x − 2x + 3y − y = 6 → 2 y = 6.
Step 3: y = 3.
Step 4: 2x + 3 = 7 → 2x = 4 → x = 2.
Step 5: 2(2) + 3(3) = 4 + 9 = 13 ✓.
Answer: (x, y) = (2, 3).
3.1, x + y = 9, x − y = 3
Add: 2x = 12 → x = 6. Back-sub: y = 9 − 6 = 3. (6, 3). Check: 6 − 3 = 3 ✓.
3.2-2x + y = 11, 2x − y = 5
Add: 4x = 16 → x = 4. Back-sub: y = 11 − 2(4) = 3. (4, 3). Check: 2(4) − 3 = 5 ✓.
3.3-3x + 2y = 16, 3x − y = 10
Subtract: (3x + 2y) − (3x − y) = 16 − 10 → 3y = 6 → y = 2. Back-sub: 3x − 2 = 10 → 3x = 12 → x = 4. (4, 2). Check Eq 1: 3(4) + 2(2) = 12 + 4 = 16 ✓.
3.4, x + 2y = 8, x + y = 5
Subtract: y = 3. Back-sub: x + 3 = 5 → x = 2. (2, 3). Check: 2 + 2(3) = 8 ✓.
3.5-4x − 3y = 6, 2x + 3y = 12
Add (opposite y signs): 6x = 18 → x = 3. Back-sub: 2(3) + 3y = 12 → 3y = 6 → y = 2. (3, 2). Check Eq 1: 4(3) − 3(2) = 6 ✓.
3.6-5x + 3y = 21, 2x + 3y = 12
Subtract (same +3y): 3x = 9 → x = 3. Back-sub: 2(3) + 3y = 12 → 3y = 6 → y = 2. (3, 2). Check Eq 1: 5(3) + 3(2) = 15 + 6 = 21 ✓.
3.7-3x + 2y = 12, x − y = 1 (×2)
Multiply Eq 2 by 2: 2x − 2y = 2. Add to Eq 1: 5x = 14 → x = 14/5. Back-sub into Eq 2: 14/5 − y = 1 → y = 14/5 − 1 = 9/5. (14/5, 9/5). Check Eq 1: 3(14/5) + 2(9/5) = 42/5 + 18/5 = 60/5 = 12 ✓.
3.8, x + 4y = 14, 3x − 2y = 0 (×2)
Multiply Eq 2 by 2: 6x − 4y = 0. Add to Eq 1: 7x = 14 → x = 2. Back-sub into Eq 2 (original): 3(2) − 2y = 0 → 2y = 6 → y = 3. (2, 3). Check Eq 1: 2 + 4(3) = 14 ✓.