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3.1 Algebraic solution

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Algebraic solution

The algebraic method of solving simultaneous equations is by substitution.

For example the solution of

y - 2 x = - 4 x 2 + y = 4

is:

y = 2 x - 4 into second equation x 2 + ( 2 x - 4 ) = 4 x 2 + 2 x - 8 = 0 Factorise to get : ( x + 4 ) ( x - 2 ) = 0 ∴ the 2 solutions for x are : x = - 4 and x = 2

The corresponding solutions for y are obtained by substitution of the x -values into the first equation

y = 2 ( - 4 ) - 4 = - 12 for x = - 4 and : y = 2 ( 2 ) - 4 = 0 for x = 2

As expected, these solutions are identical to those obtained by the graphical solution.

Solve algebraically:

y - x 2 + 9 = 0 y + 3 x - 9 = 0
  1. y + 3 x - 9 = 0 y = - 3 x + 9
  2. ( - 3 x + 9 ) - x 2 + 9 = 0 x 2 + 3 x - 18 = 0 Factorise to get : ( x + 6 ) ( x - 3 ) = 0 ∴ the 2 solutions or x are : x = - 6 and x = 3
  3. y = - 3 ( - 6 ) + 9 = 27 for x = - 6 and : y = - 3 ( 3 ) + 9 = 0 for x = 3

  4. The first solution is x = - 6 and y = 27 . The second solution is x = 3 and y = 0 .

Algebraic solution

Solve the following systems of equations algebraically. Leave your answer in surd form, where appropriate.

1. a + b = 5 a - b 2 + 3 b - 5 = 0
2. a - b + 1 = 0 a - b 2 + 5 b - 6 = 0
3. a - ( 2 b + 2 ) 4 = 0 a - 2 b 2 + 3 b + 5 = 0
4. a + 2 b - 4 = 0 a - 2 b 2 - 5 b + 3 = 0
5. a - 2 + 3 b = 0 a - 9 + b 2 = 0
6. a - b - 5 = 0 a - b 2 = 0
7. a - b - 4 = 0 a + 2 b 2 - 12 = 0
8. a + b - 9 = 0 a + b 2 - 18 = 0
9. a - 3 b + 5 = 0 a + b 2 - 4 b = 0
10. a + b - 5 = 0 a - b 2 + 1 = 0
11. a - 2 b - 3 = 0 a - 3 b 2 + 4 = 0
12. a - 2 b = 0 a - b 2 - 2 b + 3 = 0
13. a - 3 b = 0 a - b 2 + 4 = 0
14. a - 2 b - 10 = 0 a - b 2 - 5 b = 0
15. a - 3 b - 1 = 0 a - 2 b 2 - b + 3 = 0
16. a - 3 b + 1 = 0 a - b 2 = 0
17. a + 6 b - 5 = 0 a - b 2 - 8 = 0
18. a - 2 b + 1 = 0 a - 2 b 2 - 12 b + 4 = 0
19. 2 a + b - 2 = 0 8 a + b 2 - 8 = 0
20. a + 4 b - 19 = 0 8 a + 5 b 2 - 101 = 0
21. a + 4 b - 18 = 0 2 a + 5 b 2 - 57 = 0
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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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