Last reviewed: 7 November 2026
Updated November 2026. These Grade 8 linear equations notes show how to form and solve a pair of equations in two unknowns by substitution and by elimination. In substitution you make one letter the subject of one equation and put it into the other. In elimination you add or subtract the equations so that one letter disappears.
All examples are original practice for Kenyan junior school learners. For the other topics, see our Grade 8 mathematics revision guide and the Grade 8 notes hub.
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What the Grade 8 Curriculum Covers
Linear equations is a sub-strand under Algebra in the Kenya Institute of Curriculum Development’s Grade 8 Mathematics design. Learners form linear equations in two unknowns from real-life situations and solve them by substitution and elimination. The design is listed on the KICD Grade Eight curriculum designs page. Both methods are also explained in OpenStax Elementary Algebra 2e.
Forming Equations in Two Unknowns
Choose a letter for each unknown, then write one equation for each fact. Example. Two pencils and one rubber cost KSh 50. One pencil and one rubber cost KSh 35. Let a pencil cost p and a rubber cost r:
- 2p + r = 50
- p + r = 35
Solving by Substitution
- Make one letter the subject of one equation.
- Substitute that expression into the other equation and solve.
- Put the value back to find the second letter.
- Check both equations.
From p + r = 35, r = 35 − p. Substitute: 2p + (35 − p) = 50, so p + 35 = 50 and p = 15. Then r = 35 − 15 = 20. Check: 2(15) + 20 = 50. A pencil costs KSh 15 and a rubber KSh 20.
Solving by Elimination
- Make the number in front of one letter the same in both equations (multiply if needed).
- Subtract if the signs are the same; add if they are different.
- Solve for the remaining letter, then substitute back.
Example. Solve 3x + 2y = 16 and x − 2y = 0.
The y terms have opposite signs, so add: 4x = 16, x = 4. Then 4 − 2y = 0, so y = 2. Check: 3(4) + 2(2) = 16.
Example. Solve 2x + 3y = 13 and 3x + 2y = 12.
Multiply the first by 3 and the second by 2: 6x + 9y = 39 and 6x + 4y = 24. Subtract: 5y = 15, y = 3. Then 2x + 9 = 13, x = 2. Check: 3(2) + 2(3) = 12.
Which Method to Choose
Use substitution when one letter already stands alone or has 1 in front of it. Use elimination when both equations are in the form ax + by = c. Both give the same answer.
Original Practice Questions With Worked Answers
- Solve x + y = 10 and x − y = 4.
Answer: add: 2x = 14, x = 7, y = 3. - Solve y = 2x and 3x + y = 25 by substitution.
Answer: 3x + 2x = 25, x = 5, y = 10. - Solve 4x + y = 14 and 2x + y = 8.
Answer: subtract: 2x = 6, x = 3, y = 2. - Three mangoes and two oranges cost KSh 110. One mango and two oranges cost KSh 70. Find the price of each.
Answer: 3m + 2r = 110 and m + 2r = 70. Subtract: 2m = 40, m = 20. Then 20 + 2r = 70, r = 25. A mango costs KSh 20 and an orange KSh 25. - The sum of two numbers is 30 and their difference is 8. Find them.
Answer: a + b = 30, a − b = 8; 2a = 38, a = 19, b = 11.
Before this topic, revise simplifying in our Grade 8 algebraic expressions notes. Grade 9 extends equations to inequalities in our Grade 9 linear inequalities notes.
For daily practice across subjects, a Day Pass (KSh 20) or Monthly Pass (KSh 500) opens the full question bank.
Frequently Asked Questions
What is a linear equation in two unknowns?
It is an equation with two letters, each to the power 1, such as 2x + 3y = 12. You need two such equations to find both values.
When do I add and when do I subtract in elimination?
Subtract when the matching terms have the same sign. Add when they have opposite signs.
How do I check my answer?
Put both values into both original equations. If both sides are equal each time, the answer is correct.
Do substitution and elimination give different answers?
No. Both methods give the same solution; choose the one that needs less working.
Ten-minute check: try the free Grade 8 CBC Challenge. It has 10 questions, each marked on the spot with the working, and it shows which topics to practise next.