Substitution Method to solve linear equations

The substitution method is one of the techniques used to solve systems of linear equations. In this method, one variable is isolated from one of the equations and then substituted into the other equation to solve for the other variable.

Solving the given linear equations using substitution method

Given equations are:

3x/2 - 5y/3 = -2 ..(1)
x/2 + y/2 = 13/6 ..(2)

Let us solve equation (2) for one variable:

x/2 + y/2 = 13/6

x/2 = 13/6 - y/2 (subtracting y/2 from both sides)

x = (13/6 - y/2) * 2 (multiplying both sides by 2)

x = 13/3 - y ..(3)

Now, substitute the value of x from equation (3) into equation (1):

3x/2 - 5y/3 = -2

3(13/3 - y)/2 - 5y/3 = -2 (substituting x = 13/3 - y)

13 - 3y - 10y/3 = -6 (multiplying both sides by 2/3)

39 - 9y - 10y = -18 (multiplying both sides by 3)

-19y = -21

y = 21/19 (dividing both sides by -19)

Now, substitute the value of y into equation (3) to find x:

x = 13/3 - y

x = 13/3 - 21/19

x = 38/19

Therefore, the solution to the system of linear equations is:

x = 38/19, y = 21/19

Verification of the solution

Substitute the values of x and y into the original equations and check if they satisfy:

3x/2 - 5y/3 = -2

3(38/19)/2 - 5(21/19)/3 = -2

6 - 35/19 = -2

-2 = -2

x/2 + y/2 = 13/6

38/19/2 + 21/19/2 = 13/6

19/19 + 21/19 = 13/6

1 = 1

Hence, the solution (x = 38/19, y = 21/19) is verified.