Grade 8 Mathematics Notes: Linear Equations in Two Unknowns (Substitution and Elimination)

Grade 8 Mathematics · Strand 2.0 Algebra · Sub-strand 2.2 Linear Equations in Two Unknowns

These are free study notes for Grade 8 learners, written to the KICD curriculum design and taken from our full Grade 8 Mathematics notes. Word problems are where most marks are lost: learners can solve the equations but struggle to form them. Pay special attention to the sections on forming equations and shopping problems.

When two facts involve the same two unknowns, such as the prices of books and pens, we can form two equations and solve them together. In this topic we form linear equations in two unknowns and solve them by substitution and by elimination.

Forming simultaneous equations

Let a book cost b shillings and a pen cost p shillings. Two books and three pens cost Ksh 190: 2b + 3p = 190. One equation has many possible answers, so a second fact is needed: one book and two pens cost Ksh 110, so b + 2p = 110.

Substitution method

Elimination method

Shopping problems

Other real-life problems

Solving any word problem

1. Choose one letter for each unknown.
2. Write one equation for each fact.
3. Solve by substitution or elimination.
4. Check the answers in both equations and in the story.

Test yourself: multiple choice

1. Solve x + y = 20 and x - y = 6.

   1. x = 13, y = 7
   2. x = 7, y = 13
   3. x = 14, y = 6
   4. x = 10, y = 10
   Show the answer

   A: Add: 2x = 26, x = 13; y = 7.

2. Solve 5a + 2b = 24 and 3a + 2b = 16. What is a?

   1. 2
   2. 8
   3. 3
   4. 4
   Show the answer

   D: Subtract: 2a = 8, a = 4.

3. Solve y = 3x - 1 and 2x + 3y = 19. What is x?

   1. 5
   2. 1
   3. 2
   4. 3
   Show the answer

   C: 2x + 9x - 3 = 19, 11x = 22, x = 2.

4. A farm has 30 heads and 80 legs of hens and goats. How many goats?

   1. 20
   2. 10
   3. 15
   4. 5
   Show the answer

   B: h + g = 30, 2h + 4g = 80 gives g = 10.

5. To eliminate y from 3x + y = 10 and x + 2y = 5, first multiply the first equation by

   1. 2
   2. 3
   3. 5
   4. 10
   Show the answer

   A: Then both have 2y.

Practice questions

1. Solve by substitution: y = 2x + 1 and 3x + y = 21. (3 marks)

   Show the answer

   • 3x + 2x + 1 = 21
   • 5x = 20, x = 4
   • y = 9

2. Solve by elimination: 4m + 3n = 26 and 2m - 3n = -2. (3 marks)

   Show the answer

   • Add: 6m = 24, m = 4
   • 16 + 3n = 26, n = 10/3 = 3 1/3
   • Check: 8 - 10 = -2

3. Two adult tickets and three child tickets cost Ksh 700. One adult and one child ticket cost Ksh 300. Find the price of each. (4 marks)

   Show the answer

   • 2a + 3c = 700; a + c = 300
   • 2a + 2c = 600; subtract: c = 100
   • a = 200
   • Adult Ksh 200, child Ksh 100

4. The sum of two numbers is 45 and their difference is 11. Find the numbers. (3 marks)

   Show the answer

   x + y = 45, x - y = 11; 2x = 56, x = 28, y = 17.