Question

Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x and/or y.)

3x − 10y = 8
12x − 40y = 32
(x, y) =

Answer 1

Using the Gauss-Jordan row reduction method, the solution to the system is (x, y) =
`\left( x, (3)/(10)x - (4)/(5) \right) \)`, where x can be any real number.

How to use the Gauss-Jordan row reduction method?

To solve the given system of equations using Gauss-Jordan row reduction, we can represent the augmented matrix and perform row operations until we reach the reduced row-echelon form (RREF). The augmented matrix for the given system is:

`\[ \begin{bmatrix} 3 & -10 & | & 8 \\ 12 & -40 & | & 32 \end{bmatrix} \]`

Now, let's perform row operations to get the RREF:

1. Replace R2 with R2 - 4R1:

`\[ \begin{bmatrix} 3 & -10 & | & 8 \\ 0 & 0 & | & 0 \end{bmatrix} \]`

This implies that the system is dependent, and we have one equation:

3x - 10y = 8

Now, express the solution in terms of parameters. Let's solve for y in terms of x:

3x - 10y = 8

-10y = -3x + 8

y = 3/10x - 4/5

So, the solution to the system is (x, y) =
`\left( x, (3)/(10)x - (4)/(5) \right) \)`, where x can be any real number.