Simultaneous Linear Equations –
As we all know, from the previous lesson about linear equation and inequation, now we will learn about Simultaneous Linear Equations.
A linear equation in two variables (say, x & y) contains the variables in the first degree and in separate terms. The general form of such an equation is ax + by + 2 = 0, where a, b, & c are real numbers and a & b are nonzero numbers.
y
Examples.) 2x – 5y = 7 & 5x + ------ + 3 = 0 are linear equations in
2
’x’ & ‘y’ , however 5xy = 9 is not a
linear equation because ‘x’ & ‘y’ are not contained in separate terms.
Simultaneous Linear Equations –
If two linear equations in ‘x’ & ‘y’ are satisfied by the same values of ‘x’ & ‘y’ then the equations are called simultaneous linear equations. The general form of such equations is ax + by + c = 0 and px + qy + r = 0.
Example.) The equations x + 2y = 5 and 2x + y = 4 are satisfied by the values x = 1, y =2. Therefore, x + 2y = 5 and 2x + y = 4 are simultaneous linear equations and their solution is x = 1, y = 2.
To find a solution to simultaneous linear equations, we must find a pair of values of the variables that satisfy both the equations. There are two ways of doing this –
1) By Substitution 2) By Elimination
Substitution Method –
There are some steps of substitution method are given below –
Step.1) Using one of the equations write ‘y’ in terms of ‘x’ or ‘x’ in terms of ‘y’ and the constant.
Step.2) Substitute the expression for ‘y’ or ‘x’ in the second equation.
Step.3) Solve the resulting linear equations in ‘x’ or ‘y’.
Step.4) Substitute the value of ‘x’ or ‘y’ in either of the equations
Step.5) Solve the resulting linear equations in ‘y’ or ‘x’
Step.6) Verify the correctness of the solution by substituting the values of ‘x’ & ‘y’ in the given equations.
Elimination Method –
This method is also called the addition subtraction method
Step.1) Decide which variable will be easier to eliminate, try to avoid fractions
Step.2) Multiply one or both the equations by suitable numbers to ensure that the coefficients of the variable to be eliminated are the same in both the equations.
Step.3) Add or subtract the resulting equations to eliminate the variable
Step.4) Solve the resulting equation in one variable
Step.5) Substitute the value of the variable obtained in step. 4in either of the given equations.
Step.6) Solve the resulting equation.
Step.7) Verify the correctness of the solution by substituting the values of the variables in the given equations.