Solving for Ordered Pair Solutions

Introduction 

An ordered pair represents a location (object or point) in the given plane. Ordered pair consists of two terms ‘x’ and ‘y’, represented in the form (x, y). Ordered pairs are also called as co-ordinates

If only one equation is given,
ax+by=c (Here a,b and c are constants)

Then substitute different values for ‘x’ and obtain ‘y’. Therefore different ordered pairs are the different 'x' values and their                        corresponding 'y' values (x, y).

If more than one equation is given,
a1 x+ b1 y=c1

a2 x+b2 y=c2 (Here a1,a2,b1,b2,c1 and c2 are constants)

Solve those equations either by elimination or substitution or  graphing method and findout the values of ‘x’ and ‘y’. Then (x, y) is the ordered pair.

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Solving for Ordered Pair Solutions of a Linear Equation:

Ex 1:  Find the ordered pairs of the equation

x+y=1

Sol :  Subtract x on both sides, so we get y=1-x

Substitute x=0

y =1-0

y =1

Therefore the ordered pair (x, y) is (0, 1).

Substitute x=1

y =1-1

y =0

Therefore the ordered pair (x, y) is (1, 0).

Substitute x=2

y =1-2

y =-1

Therefore the ordered pair (x, y) is (2, -1).

Substitute x=3

y =1-3

y =-2

Therefore the ordered pair (x, y) is (3, -2).

The ordered pairs are (0, 1), (1, 0), (2, -1), (3, -2).

Ex 2:  Find the ordered pairs of the equation

y-5x=1

Sol :   y-5x=1,

Add 5x on both sides, so we get y =1+5x,

Substitute x=0

y =1+5(0),

y =1

Therefore the ordered pair (x, y) is (0, 1).

Substitute x = 1

y =1+5(1),

y =6

Therefore the ordered pair (x, y) is (1, 6).

Substitute x=2

y =1+5(2),

y =11

Therefore the ordered pair (x, y) is (2, 11).

Substitute x=3

y =1+5(3),

y =16

Therefore the ordered pair (x, y) is (3, 16).

The ordered pairs are (0, 1), (1, 6), (2, 11), (3, 16).

Solving for Ordered Pair Solutions of Two Linear Equations:

Ex 1 :  Find the ordered pair of the given system of equation:

7x – 15y = 2-----------Equation (1)

x + 2y = 3---------------Equation (2)

Sol  :   Step 1 : We pick either of the equations and write one variable in terms of the other.

Let us consider the Equation (2) :

x + 2y = 3

and write it as x = 3 – 2y

Step 2 : Substitute the value of x in Equation (1). We get

7(3 – 2y) – 15y = 2

i.e., 21 – 14y – 15y = 2

i.e., – 29y = –19

Therefore, y =19/29

Step 3 : Substituting this value of y in Equation (3), we get

x=3-2(19/29)=49/29

Therefor the solution is x=49/29 and y=19/29.

Verification : Substituting x =49/29 and y =19/29, we can verify that both the Equations

(1) and (2) are satisfied.