Linear Equations in Two Variables
Definition
Any equation which can be put in the form ax+by+c=0, where a,b,c are real numbers and a,b are both not zero, is called a linear equation in two variables.
Ex. 2x+4y=18,
x+y=20
Note: The general form of a linear equation in two variable is ax+by+c=0
Exercise
1) The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement
solution:
let the cost of a notebook to be Rs x and
cost of a pen to be Rs y.
The linear equation is
x=2y
or, x-2y=0
2)Express the following equations in the form ax+by+c=0 and indicate the value of a,b and c
(i) -2x+3y=6
-2x+3y-6=0
a= -2, b=3, c= -6
(ii) 5=2x
2x-5=0
a=2, b=0, c= -5
Solution of a Linear Equation
Important points
- A linear equation in two variables has infinitely many solutions
- An easy way of getting a solution is to take x=0 and get the corresponding value of y. Then put y=0 and get the value of x.
- the solution of a linear equation is not affected when (i) the same number is added to (or subtracted from) both sides of the equation (ii) multiply or divide both the sides of the equation by the same non zero number.
- The solution of the linear equation in two variable is written as ordered pair (x,y)
Exercise
(1) Find the value of k, if x=2, y=1 is a solution of the equation 2x+3y=k
Answer
Given that x=2, y=1 is a solution of 2x+3y=k
Therefore 2*2 + 3*1= k
4+3 = k
7= k
k=7
(2) Find four solutions of 2x+y=7
2x+y=7
Take x=0,
2*0 + y =7
0+y=7
y=7
(0,7) is a solution
Take y=0,
2x+0=7
2x=7
x=7/2
(7/2,0) is a solution
Take x=1,
2*1+y=7
2+y=7
y= 7-2
y=5
(1,5) is a solution
Take y=1,
2x+1=7
2x=7-1
2x=6
x=6/2
x=3
(3,1) is a solution
Home Work
1) Find the value of k, if x=1, y=5 is solution of 25x+10y=k
2) Is (3,0) is a solution of the equation 5x+2y=15.
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