POINTS TO NOTE
The solution of a linear equation is not affected when
1)the same number is either added or subtracted from both sides of the equation.
2) the same non-zero number is multiplied or divided on both sides of the equation.
Linear Equation in Two Variables
It is an equation of the form ax + by + c = 0, where a and b are real numbers, a ¹ 0, b ¹ 0 and c are constants (real numbers) and x, y are variables. Let us look at the following linear equation in 2 variables
1) x + 2y = 3,
Here x + 2y = 3 can be written as
x + 2y - 3 = 0. Here it is in the form ax + by + c = 0 where a =1, b =2 and c = -3
2) 2x - 3y - 5 = 0
It is in the form ax + by + c = 0 where a =2, b =-3 and c = -5
3) 5x+7y = 4,
Here 5x + 7y = 4 can be written as
5 x + 7y - 4 = 0
It is in the form ax + by + c = 0 where a =5, b =7and c = -4
Solution
A pair of values of x and y satisfying the equation ax + by + c = 0 is known as the solution of the equation.
Example : x = 1, y = 2 is a solution of x + 3y = 7.
Solution of a Linear Equation
We observed that every linear equation in one variable has a unique solution. The solution of a linear equation in two variables is a pair of values (x, y) which satisfy the given equation. Consider an equation 2x + 5y = 17. Here, x = 1 and y = 3 is a solution as when we substitute x = 1 and y = 3 in the equation, we get2x + 5y = 2´1 + 5´ 3 = 2 + 15 = 17
Thus the solution is (1, 3). Similarly (6, 1) is also a solution for the above equation. Try the pair (0, 3). L.H.S = 2´0 + 5´3 = 0 + 15 = 15, which is not 17. Hence (0, 3) is not a solution to the above equation. But this doesn’t mean that there are no more solutions. We can get many solutions in the following way:
Example :
1) Find four solutions of the equation x + 3y = 7
Let us choose x = 1 then 3y =6 and hence y =2
Therefore x = 1 and y =2 is a solution of the equation x + 3y = 7
If we take x =4 then 3y =3 or y =1
Therefore x = 4 and y =1 is a solution of the equation x + 3y = 7
If we assume x =0 then y = 7/3,is a solution of the equation x + 3y = 7
If we assume y = 0 then x = 7, is a solution of the equation x + 3y = 7
(1, 2), (4, 1), (0,7/3 ), (0 ,7) are the four solutions of the infinitely many solutions.
Summary
- An equation is an equality involving one or more unknown quantities known as variables.
- The general form of linear equations in two variable is ax + by + c = 0.
- Solution means a pair of values, one for x and other for y; where the values substituted in the given equation makes the two sides of the equation equal.
- A linear equation in two variables has infinitely many solutions.
- The geometrical representation of ax + by + c = 0 is a straight line.
- x= 0 is the equation of the y – axis.
- y= 0 is the equation of the x - axis.
- In a graph x = k is a straight line parallel to the y – axis.
- In a graph y = k is a straight line parallel to the x – axis.
- The type of equation y = mx represents a line passing through the origin.
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