How many solutions does the system of equations below have? x - 2y - z = 3 -2x + 3y - 3z = 0 2x + 2y - 3z = 1 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Write Equations: First, let's write down the system of equations: 1. x - 2y - z = 3 2. -2x + 3y - 3z = 0 3. 2x + 2y - 3z = 1Eliminate x: Now, let's add equations 1 and 2 to eliminate x: (1) + (2) gives us: x - 2y - z - 2x + 3y - 3z = 3 + 0 -x + y - 4z = 3Eliminate x Again: Next, let's add equations 1 and 3 to eliminate x: (1) + (3) gives us: x - 2y - z + 2x + 2y - 3z = 3 + 1 3x - 4z = 4Eliminate y: Now, let's multiply the new equation from step 3 by 2 and add it to the new equation from step 4 to eliminate y: 2(-x + y - 4z) + (3x - 4z) = 2(3) + 4 -2x + 2y - 8z + 3x - 4z = 6 + 4 x - 12z = 10

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How many solutions does the system of equations below have? $\newline$ $x - 2y - z = 3$ $\newline$ $-2x + 3y - 3z = 0$ $\newline$ $2x + 2y - 3z = 1$ $\newline$ 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

x - 2y - z = 3

\newline

-2x + 3y - 3z = 0

\newline

2x + 2y - 3z = 1

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Write Equations: First, let's write down the system of equations: $\newline$ $1$ . $x - 2y - z = 3$ $\newline$ $2$ . $-2x + 3y - 3z = 0$ $\newline$ $3$ . $2x + 2y - 3z = 1$
Eliminate x: Now, let's add equations $1$ and $2$ to eliminate x: $\newline$ $(1) + (2)$ gives us: $\newline$ $x - 2y - z - 2x + 3y - 3z = 3 + 0$ $\newline$ $-x + y - 4z = 3$
Eliminate x Again: Next, let's add equations $1$ and $3$ to eliminate x: $\newline$ $(1) + (3)$ gives us: $\newline$ $x - 2y - z + 2x + 2y - 3z = 3 + 1$ $\newline$ $3x - 4z = 4$
Eliminate y: Now, let's multiply the new equation from step $3$ by $2$ and add it to the new equation from step $4$ to eliminate $y$ : $\newline$ $2(-x + y - 4z) + (3x - 4z) = 2(3) + 4$ $\newline$ $-2x + 2y - 8z + 3x - 4z = 6 + 4$ $\newline$ $x - 12z = 10$