How many solutions does the system of equations below have? 3x + 3y + 2z = 15 -2x - 2y + z = 11 2x - y + z = 4 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Combine Equations to Eliminate z: First, let's try to simplify the system by adding the first and second equations to eliminate z. (3x + 3y + 2z) + (-2x - 2y + z) = 15 + 11 This simplifies to x + y + 3z = 26Combine Equations to Eliminate z Again: Now, let's add the second and third equations to eliminate z again. (-2x - 2y + z) + (2x - y + z) = 11 + 4 This simplifies to -3y + 2z = 15Create New Equations: We now have two new equations: x + y + 3z = 26 -3y + 2z = 15 Let's multiply the second equation by 3 to align the y terms with the first equation. 3(-3y + 2z) = 3(15) This gives us -9y + 6z = 45Align y Terms: Now we can add the new equation to the first one to eliminate y. (x + y + 3z) + (-9y + 6z) = 26 + 45 This simplifies to x - 8y + 9z = 71

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How many solutions does the system of equations below have? $\newline$ $3x + 3y + 2z = 15$ $\newline$ $-2x - 2y + z = 11$ $\newline$ $2x - y + z = 4$ $\newline$ 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 + 3y + 2z = 15

\newline

-2x - 2y + z = 11

\newline

2x - y + z = 4

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Combine Equations to Eliminate $z$ : First, let's try to simplify the system by adding the first and second equations to eliminate $z$ . $(3x + 3y + 2z) + (-2x - 2y + z) = 15 + 11$ This simplifies to $x + y + 3z = 26$
Combine Equations to Eliminate z Again: Now, let's add the second and third equations to eliminate z again. $\newline$ $(-2x - 2y + z) + (2x - y + z) = 11 + 4$ $\newline$ This simplifies to $-3y + 2z = 15$
Create New Equations: We now have two new equations: $\newline$ $x + y + 3z = 26$ $\newline$ $-3y + 2z = 15$ $\newline$ Let's multiply the second equation by $3$ to align the $y$ terms with the first equation. $\newline$ $3(-3y + 2z) = 3(15)$ $\newline$ This gives us $-9y + 6z = 45$
Align y Terms: Now we can add the new equation to the first one to eliminate y. $\newline$ $(x + y + 3z) + (-9y + 6z) = 26 + 45$ $\newline$ This simplifies to $x - 8y + 9z = 71$