To find the value of x and y that satisfies the two equations below, mean to solve the pair of simultaneous equations. Example: x+y=4 and x-y=2.

There are three methods of solving simultaneous equations. That is:

• Method of elimination
• Method of substitution
• Graphical method

1. Method of Elimination

To eliminate one of the unknowns, we make sure that the coefficient of this unknown is the same in both equations. We eliminate either by subtracting or by adding depending on the signs in front of the unknowns.

Example: Solve the following pairs of simultaneous equations using the method of elimination.

1. X+y=10 and x-y=8
2. 2x+y=5 and 3x-2y=4

Solution

1. X+y=10 - - - - (1)

x-y=8 - - - -(2)

ÞEquation (1)-(2)

2y=2;             Þy=1

Substituting y in equation (1)

X+y=10

X+1=10

X=10-1

Þ

2.       3      2x+y=5

            2        3x-2y=4

6x+3y=15 - - - - (1)

6x-4y=8 - - - - (2)

Equation (1)-(2)

Þ7y=7;        Þy=1

Substituting y=1 in equation (1)

2x+y=5

2x+1=5

2x=5-1

2x=1

Þx=2

2. Method of Substitution

1. X+y=10 and x-y=8
2. X+y=4 and 2x+y=5

Solution

1. X+y=10 - - - - (1)

X-y=8 - - - - (2)

From equation (1),

X+y=10;       Þx=10-y - - - - (3)

Put equation (3) in (2)

Þx-y=8

10-y-y=8

10-2y=8

-2y=8-10

-2y=-2;         Þy=1

Put y in (3)

X=10-y

X=10-1

X=9   

2. X+y=4 - - - - (1)

2x+y=5 - - - - (2)

From equation (1),

X+y=4 ;         Þy=4-x - - - - (3)

Put equation (3) in (2)

2x+y=5

2x + (4-x) =5

2x+4-x=5

X+4=5

X=5-4

X=1

Put x=1 in equation (3)

Y=4-x=4-1

3. Graphical Method

To solve a pair of simultaneous equation graphically, we draw the 2 equations represented by the line on the same axis. The point of intersection of the two lines gives the solution to the equations.

NB: This method requires a graph paper to have the exact method.

Examples: Solve the following pair of simultaneous equations graphically.

1. X-2y=-4 and 3x+y=9
2. 2x+y=8 and x+3y=9

Solution

1. X-2y=-4

3x+y=9

Consider the 1^st equation

X-2y=-4

When x=0, 0-2y=-4

-2y=-4;          y=2                 (0, 2)

-2    -2

When y=0, x=-2(0)=-4

X=-4                          (-4, 0)

Consider the 2^nd equation

3x+y=9

When x=0, 3(0) + y=9

Y=9                            (0, 9)

When y=0, 3x-0=9

3x=9,             x=3                 (3, 0)

2. 2x+y=8

3x=y=9

Consider the 1^st equation

2x+y=8

When x=0, 2(0) + y=8;    Þy=8                        (0, 8)

When y=0, 3x+0=8

2x=8,             x=4                             (4, 0)

From 2^nd equation

X+3y=9

When x=0, 0+3y=9

3y=9,                        Þy=3            (0, 3)

When y=0

X+3(0)=9

X=9