How many solutions does the system of equations below have? 3x - 3y + 3z = 3 -x + y - z = -1 -2x + 2y - 2z = -2 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Simplify first equation: Divide the first equation by 3 to simplify. (3x - 3y + 3z) / 3 = 3 / 3 x - y + z = 1Simplify second equation: Notice that the second equation is already simplified. -x + y - z = -1Simplify third equation: Divide the third equation by -2 to simplify. (-2x + 2y - 2z) / -2 = -2 / -2 x - y + z = 1Compare simplified equations: Compare the simplified equations. First equation: x - y + z = 1 Second equation: -x + y - z = -1 Third equation: x - y + z = 1Eliminate variables: Add the first and second equations to eliminate variables. (x - y + z) + (-x + y - z) = 1 + (-1) 0 = 0Check for dependency: The addition of the first and second equations results in a true statement, indicating that the equations are dependent.Identify same plane: Since the third equation is identical to the first, all three equations represent the same plane in three-dimensional space.Infinite solutions: The system of equations has infinitely many solutions because all three equations describe the same plane.

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How many solutions does the system of equations below have? $\newline$ $3x - 3y + 3z = 3$ $\newline$ $-x + y - z = -1$ $\newline$ $-2x + 2y - 2z = -2$ $\newline$ Choices: $\newline$ (A)no solution $\newline$ (B)one solution $\newline$ (C)infinitely many solutions

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Algebra 2

Determine the number of solutions to a system of equations in three variables

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Q. How many solutions does the system of equations below have?

\newline

3x - 3y + 3z = 3

\newline

-x + y - z = -1

\newline

-2x + 2y - 2z = -2

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Simplify first equation: Divide the first equation by $3$ to simplify. $(3x - 3y + 3z) / 3 = 3 / 3$ $x - y + z = 1$
Simplify second equation: Notice that the second equation is already simplified. $-x + y - z = -1$
Simplify third equation: Divide the third equation by $-2$ to simplify. $\frac{-2x + 2y - 2z}{-2} = \frac{-2}{-2}$ $x - y + z = 1$
Compare simplified equations: Compare the simplified equations. $\newline$ First equation: $x - y + z = 1$ $\newline$ Second equation: $-x + y - z = -1$ $\newline$ Third equation: $x - y + z = 1$
Eliminate variables: Add the first and second equations to eliminate variables. $\newline$ $(x - y + z) + (-x + y - z) = 1 + (-1)$ $\newline$ $0 = 0$
Check for dependency: The addition of the first and second equations results in a true statement, indicating that the equations are dependent.
Identify same plane: Since the third equation is identical to the first, all three equations represent the same plane in three-dimensional space.
Infinite solutions: The system of equations has infinitely many solutions because all three equations describe the same plane.