How many solutions does the system of equations below have? -3x - 3y + 3z = 12 2x + 2y - 2z = -8 x + y - z = -4 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Multiply and Compare Equations: First, let's multiply the third equation by 2 so we can compare it with the second equation. 2(x + y - z) = 2(-4) 2x + 2y - 2z = -8Check Equality of Equations: Now, let's compare the second equation with the new equation we got from doubling the third one. 2x + 2y - 2z = -8 (from the second equation) 2x + 2y - 2z = -8 (from doubling the third equation) They are the same, which means the second and third equations are actually the same line.Check for Multiple of Equations: Next, let's check if the first equation is a multiple of the second equation. -3x - 3y + 3z = 12 can be divided by -3 to get: x + y - z = -4 This is the same as the third equation, so all three equations represent the same line.Infinite Solutions Conclusion: Since all three equations represent the same line, the system has infinitely many solutions because the lines coincide.

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How many solutions does the system of equations below have? $\newline$ $-3x - 3y + 3z = 12$ $\newline$ $2x + 2y - 2z = -8$ $\newline$ $x + y - z = -4$ $\newline$ 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 + 3z = 12

\newline

2x + 2y - 2z = -8

\newline

x + y - z = -4

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Multiply and Compare Equations: First, let's multiply the third equation by $2$ so we can compare it with the second equation. $\newline$ $2(x + y - z) = 2(-4)$ $\newline$ $2x + 2y - 2z = -8$
Check Equality of Equations: Now, let's compare the second equation with the new equation we got from doubling the third one. $\newline$ $2x + 2y - 2z = -8$ (from the second equation) $\newline$ $2x + 2y - 2z = -8$ (from doubling the third equation) $\newline$ They are the same, which means the second and third equations are actually the same line.
Check for Multiple of Equations: Next, let's check if the first equation is a multiple of the second equation. $\newline$ $-3x - 3y + 3z = 12$ can be divided by $-3$ to get: $\newline$ $x + y - z = -4$ $\newline$ This is the same as the third equation, so all three equations represent the same line.
Infinite Solutions Conclusion: Since all three equations represent the same line, the system has infinitely many solutions because the lines coincide.