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.)

2x + 18y − 4z = 24
-4x − 36y + 8z = -48
(x,y,z) = _____

Answer 1

The given system of equations has an infinite number of solutions. The equations are dependent on each other, and a parametric expression for the solutions can be provided using two parameters, t and s, to express the variables x, y, and z.

Step-by-step explanation:

To solve the system of equations given, first, we observe that the second equation is simply the first equation multiplied by -2. This means they are not independent and suggests that this system may have an infinite number of solutions or no solution. The lack of a third independent equation makes it more challenging to determine the answer immediately, but due to the proportional nature of the two equations, we can simplify.

• 2x + 18y − 4z = 24
• −4x − 36y + 8z = -48

Multiply the first equation by -2:

• (-2)(2x + 18y − 4z) = (-2)(24)

• -4x - 36y + 8z = -48

Since both equations reduce to the same, we have infinite solutions. The solutions can be expressed parametrically with free variables t and s.

An example parametric solution can be:

• x = t
• y = s
• z = 3t + 9s