How many solutions does the system of equations below have? -3x - y + z = -16 2x + y + z = 5 -x - 3y + 2z = 10 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Eliminate y by adding equations: First, let's add the first and second equations to eliminate y. (-3x - y + z) + (2x + y + z) = -16 + 5 This simplifies to: -x + 2z = -11Eliminate y again by adding equations: Now, let's add the second and third equations to eliminate y again. (2x + y + z) + (-x - 3y + 2z) = 5 + 10 This simplifies to: x - 2y + 3z = 15Eliminate x by multiplying and adding equations: We can multiply the first equation by 3 and add it to the third equation to eliminate x. 3(-3x - y + z) + (-x - 3y + 2z) = 3(-16) + 10 This simplifies to: -10x - 6y + 5z = -38 Oops, I made a mistake here. The correct calculation should be: -9x - 3y + 3z - x - 3y + 2z = -48 + 10 Which simplifies to: -10x - 6y + 5z = -38

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How many solutions does the system of equations below have? $\newline$ $-3x - y + z = -16$ $\newline$ $2x + y + z = 5$ $\newline$ $-x - 3y + 2z = 10$ $\newline$ Choices: $\newline$ (A)no solution $\newline$ (B)one solution $\newline$ (C)infinitely many solutions

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Precalculus

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 - y + z = -16

\newline

2x + y + z = 5

\newline

-x - 3y + 2z = 10

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Eliminate y by adding equations: First, let's add the first and second equations to eliminate y. $\newline$ $(-3x - y + z) + (2x + y + z) = -16 + 5$ $\newline$ This simplifies to: $\newline$ $-x + 2z = -11$
Eliminate $y$ again by adding equations: Now, let's add the second and third equations to eliminate $y$ again. $\$2x + y + z$ + (-x - $3$ y + $2$ z) = $5$ + $10$ \)This simplifies to: $x - 2y + 3z = 15$
Eliminate x by multiplying and adding equations: We can multiply the first equation by $3$ and add it to the third equation to eliminate x. $\newline$ $3(-3x - y + z) + (-x - 3y + 2z) = 3(-16) + 10$ $\newline$ This simplifies to: $\newline$ $-10x - 6y + 5z = -38$ $\newline$ Oops, I made a mistake here. The correct calculation should be: $\newline$ $-9x - 3y + 3z - x - 3y + 2z = -48 + 10$ $\newline$ Which simplifies to: $\newline$ $-10x - 6y + 5z = -38$