# How to solve the system of linear equations in a easy way?

420 Points
13 years ago

Dear Suresh

Solving systems of equations graphically is one of the easiest ways to solve systems of simple equations

Another way to solve systems of equations is by substitution.  In this method, you solve on equation for one variable, then you substitute that solution in the other equation, and solve.  Example:

```1. Problem: Solve the following system:            x + y = 11            3x - y = 5              Solution: Solve the first equation for y            (you could solve for x - it            doesn't matter).                        y = 11 - x                        Now, substitute 11 - x for y            in the second equation. This gives            the equation one variable, which            earlier algebra work has taught            you how to do.                        3x - (11 - x) = 5            3x - 11 + x = 5            4x = 16            x = 4                        Now, substitute 4 for x in            either equation and solve for y.            (We use the first equation below.)                        4 + y = 11            y = 7Solve the following system:             x + y + z  = 4             x - 2y - z = 1            2x - y - 2z = -1  Solution: Start out by multiplying the            first equation by -1 and add            it to the second equation to             eliminate x from the second            equation.            -x  - y - z = -4             x - 2y - z = 1            ----------------               -3y - 2z = -3            Now eliminate x from the third            equation by multiplying the first            equation by -2 and add it to            the third equation.            -2x - 2y - 2z = -8             2x -  y - 2z = -1            ------------------                 -3y - 4z = -9            Next, eliminate y from the third            equation by multiplying the second            equation by -1 and adding it to            the third equation.             3y +  2z = 3             -3y -  4z = -9            --------------                  -2z = -6            Solve the third equation for z.                             -2z = -6              z = 3            Substitute 3 for z in the            second equation and solve for y.            -3y - 2z = -3            -3y - 2(3) = -3            -3y - 6 = -3            -3y = 3            y = -1            Lastly, substitute -1 for y and            3 for z in the first equation            and solve for x.            x + (-1) + 3 = 4            x + 2 = 4            x = 2
All the best.
AKASH GOYAL