Introduction

Simultaneous linear equations involving two variables.

Consider the equation.

x + y =5

It has several values for x and y that can satisfy x + y =5

These are shown in the table below.

x 0 1 2 3 4 5

y 5 4 3 2 1 0

However, if a second equation relating x and y is given

x - y = 1, the solution of the equations are the values of x and y which satisfy both equations at the same time.

x 0 1 2 3 4 5

y -1 0 1 2 3 4

x - y 1 1 1 1 1 1

The values x = 3 and y = 2 satisfies both the equations x + y = 5, and x - y = 1.

To solve an equation with two variables, we require at least two equations relating to the two variables. These are called Simultaneous equations.

Solving simultaneous equations by elimination method.

. 3x - 2y = 11

-(2x - 2y = 10)

Method

Let 3x - 2y = 11 be equation (i

and 2x - 2y = 10 be equation (ii)

Substract equation (ii) from equation (i)

3x - 2y = 11

- 2x - 2y =10

x = 1

The Value of x is now equal to 1. In equation (i) or (ii) substitute 1 for x.

Take3x - 2y = 11

3x1 - 2y = 11

1 - 2y = 11

3 - 3 - 2y = 11 - 3

-2y = 8

__ __

-2 -2

y = -4

The values of x and y that satisfies both simultaneous linear equations are x = 1 and y = -4.

Solve the simultaneous equation

2x - 4y = 2

+(12x + 4y = 40)

Let 2x - 4y = 2 be equation (i)

and 12x + 4y = 40 be equation (ii)

Add equation (i) to equation (ii)

2x - 4y = 2

+(12x + 4y = 40)

_____

14x = 42

Solve this linear equation by dividing

both sides of the equation by 14.

14x = 42

____

14 14

x =3

The value of x is 3.

In equation (i) and (ii) substitute 3 for x.

Take 2x - 4y = 2

2x3 - 4y = 2

2 - 4y = 2

3 - 6 - 4y = 2 -6

-4y = -4

-4y= -4

y = 1

The values of x and y that

satisfies both simultaneous

linear equations are x = 3

and y = 1

In the above two illustrations, it was easy to eliminate one unknown for its coefficients were numerically equal. When the signs infront of the unknown to be eliminated are the same we subtract, and when the signs are different we add.

Solve the simultaneous equations.

9x + 3y = 21

3x - 6y = 0

Let 9x + 3y = 21 be equation (i)

and 3x - 6y =0 be equation (ii)

Note the coefficients of the unknowns are not the same.

Multiply equation (ii) by 3 to make the coefficients of x equal numerically.

(3x - 6y = 0) x3

9x - 18y = 0

Let this equation be equation (iii)

9x - 18y = 0 ----------(iii)

Subtract equation (iii) from equation (i) to eliminate x.

9x + 3y = 21 --------(i)

-(9x - 18y = 0 ---------(iii) )

______

21y = 21

Dividing both sides by 21

21y = 21

______

21 21

y = 1

In equations (i), (ii) or (iii) substitute 1 for y.

9x + 3y = 21

9x + 3 x 1 = 21

9x + 3 = 21

9x + 3 - 3 = 21 – 3

9x = 18

_______

9 9

x = 2

The values of x and y that satisfies the equations are x = 2 and y = 1

Solve the simultaneous linear equations

5x + 4y = 17

3x + 3y = 12

Let 5x + 4y = 17 be equation (i)

and 3x + 3y = 12 be equation (ii)

5x + 4y = 17 --------- (i)

3x + 3y = 12 --------- (ii)

Multiply equation (i) by 3 and equation (ii) by 5, so as to make the coefficients of x equal numerically.

3 (5x + 4y = 17)

15x + 12y = 51. Let this be equation (iii)

and 5(3x + 3y = 12)

15x + 15y = 60. Let this be equation (iv)

Subtract equation (iv) form equatiom (iii) to eliminate x now that

its coefficients are numerically equal in both equations.

15x + 12y = 51----------(iii)

-( 15x + 15y = 60 ---------- (iv))

_______

-3y = -9

_______

-3 -3

y = 3

In either equations (i), (ii), (iii) or (iv) substitute 3 for y to solve for x.

Taking 3x + 3y = 12

3x + 3(3) = 12

3x + 9 = 13

3x + 9 - 9 = 12 – 9

3x = 3

x = 1

Note in solving the simultaneous equation above, that is

15x + 12y = 51----------(iii)

-(15x + 15y = 60 ---------- (iv))

Multiply equation (i) by 3 and equation (ii) by 4 to make the coefficients of y equal numerically.

3(5x + 4y = 17)

4(4x + 3y = 12)

15x + 12y = 51 Let this be equation (iii)

12x + 12y = 48. Let this be equation (iv)

When solving simultaneous equations by eliminations method.

Decide which variable to eliminate.

Make the coefficient of the variable the same in both equations.

Eliminate the variable by addition or subtraction as is appropriate.

Solve for the remaining variable

Substitute your value for part (4) above in any of the original equations to solve for the other variable.