Question

Solve. You will find a system with an infinite number of solutions, with no solution, or with a unique solution. (Enter your answers as a comma-separated list. If there are infinitely many solutions,

enter a parametric solution using t and s as the parameters. If there is no solution, enter NONE.)
3x-21y-27z =36
-4x+28y+ 36z = -48

Answer 1

To solve the system of equations:

3x - 21y - 27z = 36

-4x + 28y + 36z = -48

We can use the method of elimination to simplify the equations and solve for the variables.

First, let's multiply the second equation by 3 to make the coefficients of x in both equations the same:

-12x + 84y + 108z = -144

Now, we can add the two equations together:

(3x - 21y - 27z) + (-12x + 84y + 108z) = 36 + (-144)

This simplifies to:

-9x + 63y + 81z = -108

We can see that the resulting equation is a multiple of the first equation:

-9(3x - 21y - 27z) = -108

Simplifying further:

-27x + 189y + 243z = -324

We can see that the two equations are equivalent. Therefore, the system of equations has infinitely many solutions. We can express the solution using parameters t and s as follows:

x = t

y = s

z = (324 - 189s - 27t) / 243

The system has infinitely many solutions represented by the parametric equations above.