A linear equation is an equation with the degree 1. It means, the highest power of the variable in the linear equation is 1. The simultaneous linear equations are set of linear equations with the same variables.
Example:
3x+y=1; x+y=5
Solution of Simultaneous linear equations:
The set of values of variables which satisfy each and every equation is called the solution of simultaneous linear equations.
Example:
x=-2, y=7 is the solution of the simultaneous linear equations 3x+y=1; x+y=5.
Process of finding the solution:
There are two methods in solving the simultaneous linear equations with 2 variables.
Method 1: Substitution method:
(i) Find the value of one variable in terms of the other variable from any one of the 2 given equations.
(ii) Substitute this value into the other equation (which is not the above selected equation) and here we get a linear equation with one variable.
(iii) Solve this equation to find the value of the above variable.
(iv) Substitute this value into the equation from step (i) to find the value of other variable.
Example:
Solve 3x+y=1; x+y=5.
Solution:
3x+y=1 … (1)
x+y=5 … (2)
From equation (2),
y=5-x … (3)
Substituting this into equation (1),
3x + 5 – x = 1
2x = -4
x=-2.
From equation (3),
y = 5-(-2)=5+2=7.
So the solution is x=-2 and y=7.
Method 2: Adding/subtracting the equations:
(i) Make the coefficient of one of the variables same in both equations.
(ii) Add or subtract the equations so that the above variable is eliminated and here, we get a new equation with the other variable.
(iii) This new equation equation is solved to find this second variable.
(iv) This value of the second variable is substituted in any of the given two equations to find the value of first variable.
Example:
Solve 3x+y=1; x+y=5.
Solution:
3x+y=1 … (1)
x+y=5 … (2)
The coefficient of y is already same (as 1) in both the equations.
Subtracting (2) from (1),
2x = -4
x= -2.
From (2),
-2 + y = 5
y = 5+2 = 7.
So the solution is x=-2 and y=7.
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