Question

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

⎧⎩⎨⎪⎪x−y−z=44x−4y−4z=163x−3y−3z=12
{
𝑥
−
𝑦
−
𝑧
=
4
4
𝑥
−
4
𝑦
−
4
𝑧
=
16
3
𝑥
−
3
𝑦
−
3
𝑧
=
12

(1 point)
Responses

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

Two of the planes are parallel, which means there are no solutions.
Two of the planes are parallel, which means there are no solutions.

All three planes intersect at a single point, which means there is one solution.

Answer 1

To determine the solution scenario for the given system of equations, we can analyze the equations step by step. The system of equations you provided is:

1. \( x - y - z = 4 \)
2. \( 4x - 4y - 4z = 16 \)
3. \( 3x - 3y - 3z = 12 \)

First, let's simplify the second and third equations.

**Second equation:**
\[
4x - 4y - 4z = 16
\]
Divide everything by 4:
\[
x - y - z = 4
\]
This is the same as the first equation, meaning that the second equation does not introduce a new plane; it coincides with the first plane.

**Third equation:**
\[
3x - 3y - 3z = 12
\]
Divide everything by 3:
\[
x - y - z = 4
\]
Again, this is the same as the first equation, meaning that the third equation also does not introduce a new plane; it coincides with the first plane.

Since all three equations represent the same plane, all planes overlap, leading to infinitely many solutions.

Hence, the correct response is:

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