Mathematics • Year 8 • Unit 2 • Lesson 17
Introduction to Simultaneous Equations
Two equations, two unknowns, one shared answer. Build fluency with checking solutions, reading the solution from an intersection point, classifying systems (one / none / infinite), and setting up two equations from a word problem.
1. I do, fully worked example
Read every line. The pair (x, y) is a solution of a system only if it satisfies BOTH equations. Checking is just substituting and comparing LHS with RHS.
Problem. Is (3, 2) the solution of the system below?
Eq 1: x + y = 5
Eq 2: x − y = 1
Step 1, Substitute into Eq 1: replace x with 3, y with 2.
LHS = 3 + 2 = 5. RHS = 5. LHS = RHS ✓
Reason: the pair must satisfy Eq 1, so we test it there first.
Step 2, Substitute into Eq 2 with the SAME pair.
LHS = 3 − 2 = 1. RHS = 1. LHS = RHS ✓
Reason: the pair must ALSO satisfy Eq 2. Passing both is the test.
Step 3, State the conclusion.
Both checks pass → (3, 2) IS the solution.
Graphically: the two lines y = 5 − x and y = x − 1 cross exactly at (3, 2).
Answer: Yes, (3, 2) is the solution.
2. We do, fill in the missing steps
Same shape as Section 1, with blanks. Fill every blank. 5 marks
Problem. Is (4, 1) the solution of the system below?
Eq 1: x + y = 5
Eq 2: 2x − y = 7
Step 1, Substitute into Eq 1.
LHS = ______ + ______ = ______. RHS = 5. LHS ___ RHS ( ✓ or ✗ )
Step 2, Substitute into Eq 2 with the SAME pair.
LHS = 2(______) − ______ = ______ − ______ = ______. RHS = 7. LHS ___ RHS ( ✓ or ✗ )
Step 3, Conclusion: Because the pair satisfies ______ equation(s), it ______ (is / is not) the solution of the system.
3. You do, independent practice
Show every check. 3.1–3.3 are foundation (verify pairs). 3.4–3.6 are standard (reading intersection points and classifying systems). 3.7–3.8 are extension (setting up two equations from words).
Foundation, verify the pair
3.1 Is (2, 3) a solution of x + y = 5 and y = 2x − 1? Show both checks. 2 marks
3.2 Is (5, 0) a solution of x + y = 5 and x − y = 4? 2 marks
3.3 Two lines on a graph intersect at the point (−1, 4). State the solution of the simultaneous equations they represent. 1 mark
Standard, classify the system
3.4 Lines y = 2x + 1 and y = 2x − 3 are graphed on the same axes. State how many solutions the system has and explain in one sentence. 2 marks
3.5 Lines y = x + 2 and 2y = 2x + 4 are graphed on the same axes. State how many solutions the system has and explain why. 2 marks
3.6 Lines y = 3x − 2 and y = −x + 6 have different gradients. Without graphing, state how many solutions the system has and explain in one sentence. 2 marks
Extension, set up two equations from words
3.7 At the canteen, 1 sandwich plus 2 drinks costs $9, and 1 sandwich plus 1 drink costs $7. Let s = price of a sandwich and d = price of a drink. Write the TWO equations, do not solve. 2 marks
3.8 The sum of two numbers is 14 and their difference is 4. Let the bigger number be x and the smaller be y. Write two equations relating x and y, then check that (9, 5) satisfies both. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (4, 1)
Step 1: LHS = 4 + 1 = 5. RHS = 5. LHS = RHS ( ✓ ).
Step 2: LHS = 2(4) − 1 = 8 − 1 = 7. RHS = 7. LHS = RHS ( ✓ ).
Step 3: Pair satisfies both equations, so it is the solution.
3.1, (2, 3)
Eq 1: 2 + 3 = 5 ✓. Eq 2: y = 2x − 1 → 3 = 2(2) − 1 = 3 ✓. Both pass → YES, (2, 3) is a solution.
3.2, (5, 0)
Eq 1: 5 + 0 = 5 ✓. Eq 2: 5 − 0 = 5, but RHS = 4. 5 ≠ 4 ✗. NOT a solution Eq 2 fails.
3.3, Intersection at (−1, 4)
The solution is read straight off the intersection: x = −1, y = 4.
3.4, y = 2x + 1 and y = 2x − 3
No solutions. Same gradient (m = 2) but different y-intercepts → the lines are parallel and never cross.
3.5, y = x + 2 and 2y = 2x + 4
Infinite solutions. Divide the second equation by 2: y = x + 2, the SAME line as the first. Every point on the line satisfies both equations.
3.6, y = 3x − 2 and y = −x + 6
Exactly one solution. Different gradients (3 vs −1) mean the lines must cross at exactly one point.
3.7, Canteen prices
Eq 1: s + 2d = 9. Eq 2: s + d = 7.
3.8, Sum 14, difference 4
Eq 1: x + y = 14. Eq 2: x − y = 4.
Check (9, 5): Eq 1: 9 + 5 = 14 ✓. Eq 2: 9 − 5 = 4 ✓. (9, 5) IS the solution.