Question

Determine whether the system has no solutions, one solution or infinitely manysolutions.-3x - 5y +36z = 76-3 +7z = 121+ x + y - 10z = -19Pls see the picture O No solutionsOne solutionO Infinitely many solutions

Answer 1

We have the following system of equations

`\begin{gathered} -3x-5y+36x=76 \\ -x+7z=12 \\ 1+x+y-10z=-19 \end{gathered}`

By subtracting 1 to the last equation, we have an equivalent system of equations:

`\begin{gathered} -3x-5y+36x=76 \\ -x+7z=12 \\ x+y-10z=-20 \end{gathered}`

and we need to find the solutions (or no solutions) of this system.

Solving by elimination method.

By adding the second and third equations, we have

`y-3z=-8\ldots(iv)`

so we have eliminated the variable x. Now, by multiplying by -3 the second equation

`\begin{gathered} -3x-5y+36x=76 \\ 3x-21z=-36 \\ \\ \end{gathered}`

and adding the result to the first one, we get

`-5y+15z=40\ldots(v)`

By combining equations (iv) and (v), we have the following system in 2 variables:

`\begin{gathered} y-3z=-8 \\ -5y+15z=40 \end{gathered}`

Lets eliminate the variable z. In this regard, by multiplying the first equation by 5, we have

`\begin{gathered} 5y-15z=-40 \\ -5y+15z=40 \end{gathered}`

and by adding both equations, we get

`0=0`

This means that there are dependent equations. So, we can write 2 variables in terms on one of them. Therefore, the answer is option 3: Infinitely many solutions.