Introduction:

               We are aware  that the general form of a linear equation is of the form       ax + by + c = 0, where x and y are the variables, and a, b , c are the numerical constants.   When we have pair of equations of this form we can solve them to identify if the graph of these lines are intersecting, parallel or coincide. Among the various methods of solving linear equations of two variables, let us discuss some of the methods in this section.          

How to solve linear Equations by substitution:

Example : Solve the following system of linear equations by substitution.
3x +  y = 8; 3x - 2y = 2

Solution:

                               Let  3 x + y = 8 --------------------------(1)
                                      3x - 2y = 2 --------------------------(2)
                                From (1), y = 8 - 3x
                            Substituting y = 8 - 3x in Equation (2), we get,
                          3x - 2 ( 8 - 3x ) = 2
                         => 3x - 16 + 6x = 2
                                 => 9x - 16 = 2
                          => 9x = 2 + 16 = 18
                                        =>  x =  18/9 = 2
                Substituting x = 2 in y = 8 - 3x, we get,
                                              y = 8 - 3 (2) = 8 - 6 = 2
Hence the solution is ( x, y) = ( 2, 2).

How to solve Linear Equations by Elimination:

While solving Linear equations of two variables by the method of elimination, we follow the following steps.
Step 1:Number the two equations as (1) and (2).
Step 2: Multiply the two equations by the suitable numbers so that the coefficients of one of the variable is same.
Step 3: Add or subtract the two equations so that one of the variable is elimination and we obtain the linear equation of one variable.
Step 4: Solve for the variable obtained from the equation in step 3.
Step 5: Substitute the value of this variable in any one of the two equations, so that we get a linear equations of the other variable.
Step 6 Solve for the other variable obtained in Step 5.
Step 7: Express the values of x and y obtained from Step 3 and step 6 in the form ( x, y) = (  ___ , ___  ).

Let us Discuss the following Example:
                                   3x - 5y = -11 ------------------------------------(1)
                                 4x + 3y  = 24  ------------------------------------(2)
                                                                To eliminate the variable y,  let us multiply equation (1) by 3 and multiply equation (2) by 5
                            3 ( 3x - 5y ) = 3(-11)   -------------------------------(3)
                           5 ( 4x + 3y ) = 5 ( 24 ) = 120 ----------------------(4)
                                9x - 15 y = -33   -----------------------------------(3)
                             20 x + 15 y = 120 ------------------------------------(4)
                            _____________________

                                29 x         = 87
                                            x = 87/29 = 3
                          Substituting x= 3 , in Equation (1) we get,
                               3 (3) - 5 y = -11
                                    9 - 5 y = -11
                                   => - 5y = -11 - 9 = - 20
                                  =>      y = - 20 / - 5 = + 4
Hence the final solution is ( x, y) = ( 3, 4)
Verification:
substituting the solution in equation (2), we get,
                 4x + 3y = 24
    => 4 ( 3) + 3 ( 4) = 24
            => 12 + 12 = 24
                    => 24 = 24
As we have same values on both sides of the equation, our solution is correct.

How to solve linear equations step by step

To solve linear equations of two variables the method of substitution is the basic method. Let us see how to solve linear equations step by step.

Step 1: Number the two given equations as (1) and (2).
Step 2: Find the value of one of the variable from any one of the equation in terms of other variable., say y from equation(1)(i.e, make one of the variable as subject of the formula )
Step 3: Substitute the value of y from step 2 in the equation (2), and reduce the equation (2) into the equation of single variable which will be x.
Step 4: solve for this variable x, since we have a linear equation in one variable x.
Step 5: Substitute this value of x, in the value of y, obtained from Step 2.
Step 6: Now we have the linear equation of one variable y, we can solve for y.
Step 7: Write the final solution in the form (x,y) = (  ___, ___ )

Linear Equations Word Problems:

Let us discuss some of the word problems. While solving word problems we should proceed the following steps.
Step 1:  Read the problem carefully and and assume the the unknowns as x and y respectively.
Step 2: Frame the equations according to the given condition and write it in the the general form ax + by + c = 0.
Step 3: Solve the two equations for the variables x and y by Substitution method or Elimination Method.
Step 4: Write the final answer of the question as given in the problem from Step 3.
Let us discuss a of word problem.
Example 1: The sum of two numbers is 69 and their difference is 17. Find the numbers
Solution: Let the two numbers  are x and y.
Sum of the two numbers = x+ y = 69  -------------------(1)
Difference of two numbers = x - y = 17 ------------------(2)
Let us solve by Substitution Method.
Let us find y from Equation (1),
                              x + y = 69 => y = 69 - x
Substituting y = 69 - x in equation (2), we get,
                            x - y = 17
              =>          x - ( 69 - x) = 17
             =>           x - 69 +   x  = 17
              =>          x + x          = 17 + 69
              =>                      2x = 86
             =>                         x = 86/2 = 43
Substituting x = 43 in Equation (2), we get,
                                      x - y = 17
                               =>   43 - y = 17
                              =>        -y = 17 - 43
                               =>       -y = - 26
                                    =>   y = 26 [ multiplying by -1 on both sides ]
               We have got x = 43 and y = 26
               Hence the two numbers are 43 and 26

Practice Questions:

1. Solve the linear equations by Substitution.
  x + y = -1 ; y - 2x = -4

2. Solve the linear equations by Elimination.
 4x - 5y = -4, 3x - 2y = -3

3. The monthly incomes of Pinky and Peyton are in the ratio 7 : 5 and their expenditures are in the ratio 3 : 2. If each saves 1500 dollars per month, find their monthly incomes.

*****