Question

sketch the graph or use a graphing utility to determine which of the following solution scenarios is true for the system of equations.

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(1 point)
all three planes intersect at a single point, which means there is one solution.
two of the planes are parallel, which means there are no solutions.
all three planes are parallel, which means there are no solutions.
all three planes overlap, which means there are infinite solutions.

Explanation:

Step1: Simplify the second and third equations

Divide the second equation \(4x - 4y - 4z=16\) by \(4\), we get \(x - y - z = 4\).
Divide the third equation \(3x - 3y - 3z=12\) by \(3\), we get \(x - y - z = 4\).

Step2: Analyze the relationship between the equations

The first equation is \(x - y - z = 4\), the second - simplified equation is \(x - y - z = 4\), and the third - simplified equation is \(x - y - z = 4\). So all three equations represent the same plane.

Answer:

All three planes overlap, which means there are infinite solutions.