Question

How do you solve using gaussian elimination or gauss Jordan elimination, 2x + 6y = 16, 2x + 3y = -7 ?

Answer 1

We try to eliminate variables to solve the equation in the gaussian elimination method, first we write all the coefficients and constants of the equation in a matrix and then we perform row operation to eliminate variables. If a system of equation contains equation ax + by =c and ex + fy = g the matrix will be $\left| \begin{matrix} a & b & c \\\ e & f & g \\\ \end{matrix} \right|$

Complete step by step solution:
Equations given in the question is 2x + 6y = 16 and 2x + 3y = -7, if we write in matrix form, we get
$\left| \begin{matrix} 2 & 6 & 16 \\\ 2 & 3 & -7 \\\ \end{matrix} \right|$
Now we can perform row operation ${{R}_{2}}$, subtracting ${{R}_{1}}$ form ${{R}_{2}}$ we get
$\left| \begin{matrix} 2 & 6 & 16 \\\ 0 & -3 & -23 \\\ \end{matrix} \right|$
Now we can say -3y is equal to -23, so y is equal to $\dfrac{23}{3}$
If we put y equal to $\dfrac{23}{3}$ in 2x + 6y = 16 we get 2x + 46 = 16 so x is equal to -15

So x = -15 and y = $\dfrac{23}{3}$ are solution of the system of equation.

Note: Given a system of equations contains n variable and n equations. We form a matrix by taking only coefficients of the variables in the equation. So it will be a $n\times n$ matrix. If the determinant value of the matrix is equal to a non-zero number then we can be sure that there will be only one solution to the system. If the determinant value will be 0 then, it will become inconsistent that means there may not exist any solution or there may be infinite solution to the system.