SIMULTANEOUS EQUATIONS PROOF AND THEOREMS

CHAPTER ONE

SIMULTANEOUS EQUATIONS

DEFINITION:

When two or more equation are satisfied by the same set of value of the unknown parameters, at the same time, they are say that they are satisfied simultaneously i.e at the same time.

Example: The set x 3. y1 satisfied the equations 2x y 7 and 3x 2y time. And we say that x 3 y1 is the simultaneous solution of the above equation.

Method of solution of simultaneous equations:

1Method of Substitution: This should be used be used when one of the coefficients of the variables is one of the two equation is unity.

e.g. 1: Solve the equation 2x y 7 1 and 3x 2y 7 2

Solution 2x y 7 1 acknowledge receipt

3x 2y 7 1

From 1 y 7 2x 3

Substitute for y in 2 to have 3x 27 2x 7 i.e.

X 3 substitute for the value of x in 3 to have

Y 7 22 x 3 7 1: x 3, y 1

Example 2

Find the value of x and y if x y 1 0 and x 2y 7 2

Solution: x y 1 1

x 2y 7 2

From 1 x 1 y 3

Substitute for x in 2 to have 1 y 2y 7 i.e . 3y 7 1 6 or y2

Substitute for this value of y in 3 to have x 2 1 3

: x 3 and y 2.

2 Method of Elimination: This consists of getting rid of one letter by making its coefficients equal in both equations and then adding or subtracting as may be necessary when the coefficients of the letters are larger. This method is generally used.

E.g: Solve the equations 3x 2y 12 and 5x 3y 1

Solution: 3x 2y 12 1

5x 3y 1 2

We eliminate y say 1 x 5 9x 6y 36 3

2 x2 10x 6y 2 4

3 4 16x 38

X 2

Substitute for x in 1 to have 3x2 2y 12

2y 126 i.e. y 3

:. x 2, y 3.

For what values of n and n are the equations 4m 6n 136 and 6m 5n 164 simultaneously satisfied

Solution: 4m 6n 136 1

6m 5n 164 2

We eliminate m say

1 x 3 gives 19m 18n 403 3

2 x 2 gives 12m 10 320 4

8n 80

Substitute for n in 4 to have 4m 136 60 76

:. M 19, n 10

Problems Leading to Simultaneous Equation:

When there are two or more unknown quantities to be found, tow or more sets of facts must be given, from which two or more equations can be written down and solved.

E.g 1: Mary went to the market and bought four knives and six forks from the one shop and paid N136 together. She went to another shop and bought 6 knives and 5 forks and paid N164 together. What was the cost of a knife and a fork