Question

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x and y in terms of the parameter t.) 3x + 6y = 15 −3x − 6y = −15

Answer 1

The system of linear equations given has an infinite number of solutions. By using Gaussian elimination, we deduced that any value of x and y satisfying one equation will satisfy the other. Thus, the solution is represented parametrically as x = t and y = 2.5 - 0.5t.

Step-by-step explanation:

The student is asked to solve a system of linear equations using methods such as Gaussian elimination or Gauss-Jordan elimination. The system given is:

3x + 6y = 15

−3x − 6y = −15

To solve the system using Gaussian elimination, we can add the two equations together:

3x + 6y = 15

−3x − 6y = −15

-----------------

0x + 0y = 0

With the addition of these two equations, we see that the left side reduces to 0 and the right side is also 0. This means that any value for x and y that satisfies one equation will satisfy the other. Therefore, this system has an infinite number of solutions.

Since we need to express x and y in terms of the parameter t, we will assume: x = t

Substitute x = t into the first equation:

3t + 6y = 15

Now, solve for y:

6y = 15 - 3t

y = ⅔(15 - 3t)

y = ⅔ * 15 - ⅔ * 3t

y = ⅔2.5 - 0.5t

The solutions can be expressed as: x = t, y = 2.5 - 0.5t.

Answer 2

`x=15-2t\,,\,y=t`

Step-by-step explanation:

Matrix is a rectangular array in which elements are arranged in rows and columns.

If number of rows is m and number of columns is n, then order of the matrix is
`m* n`.

Given:

`3x + 6y = 15\\-3x -6y = -15`

Consider
`\begin{pmatrix} 3&6&15\\-3&-6&-15\end{pmatrix}`

Apply row operation:
`R_2\rightarrow R_2+R_1`

`\begin{pmatrix} 3&6&15\\0&0&0\end{pmatrix}`

Apply row operation:
`R_1\rightarrow (R_1)/(3)`

`\begin{pmatrix} 1&2&5\\0&0&0\end{pmatrix}`

So, we get
`x+2y=15`

Take
`y=t\Rightarrow x=15-2y=15-2t`

So,
`x=15-2t\,,\,y=t`

Here, for different values of the parameter t, we get different values of x and y.

So, the system of equations has infinite solutions.