# How do you find the values of x and y given [(27),(8)]=[(3y), (5x-3y)]?

Dec 30, 2016

#### Explanation:

The Coefficient Matrix is:

[ (0,3), (5,-3) ]

Before one begins to solve a system of equations, it is best to be sure that the Determinant of the coefficient matrix is not zero:

| (0,3), (5,-3) | = (0)(-3) - (5)(3) = -15

The determinant is not zero, therefore, the system of equations has a unique solution.

You can write the system using the coefficient matrix multiplied by a Column Vector of unknown variables that is equal equal to a column vector of constants as follows:

[ (0,3), (5,-3) ] [(x), (y)] = [(27),(8)]

And then perform Elementary Row Operations

However, I recommend that you eliminate the column vector of unknowns, and merge the coefficient matrix with the column vector of constants into an Augmented matrix

[ (0,3,|,27), (5,-3,|,8) ]

And then perform Elementary Row Operations

${R}_{1} \leftrightarrow {R}_{2}$

[ (5,-3,|,8), (0,3,|,27) ]

${R}_{1} + {R}_{2} \to {R}_{1}$

[ (5,0,|,35), (0,3,|,27) ]

$\left(\frac{1}{5}\right) {R}_{1}$

[ (1,0,|,7), (0,3,|,27) ]

$\left(\frac{1}{3}\right) {R}_{2}$

[ (1,0,|,7), (0,1,|,9) ]

Read the solutions from the right side of the matrix :

$x = 7 , \mathmr{and} y = 9$

Check

$3 \left(9\right) = 27$
$5 \left(7\right) - 3 \left(9\right) = 8$

$27 = 27$
$8 = 8$

This checks.