Question

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

Answer 1

Write an Augmented matrix and then perform Elementary Row Operations

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 harr R_2`

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

`R_1 + R_2 to R_1`

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

`(1/5)R_1`

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

`(1/3)R_2`

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

Read the solutions from the right side of the matrix :

`x = 7, and y = 9`

Check

`3(9) = 27`
`5(7) - 3(9) = 8`

`27 = 27`
`8 = 8`

This checks.