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

Multiply and Compare Equations: First, let's multiply the second equation by 2 to see if it matches any of the other equations. 2(-x + y - z) = 2(5) -2x + 2y - 2z = 10Analyze Equations: Now, we compare this new equation with the first and the third equations. The first equation is -2x + 2y - 2z = 10, which is the same as the new equation we got from doubling the second one. The third equation is 2x - 2y + 2z = -10, which is the exact opposite of the first equation.Determine Solution Type: Since the first and the third equations are opposites, adding them will result in 0 = 0, which is always true. And since the second equation is a multiple of the first, it's not independent. This means the system has infinitely many solutions because the equations are dependent.

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

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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

-2x + 2y - 2z = 10

\newline

-x + y - z = 5

\newline

2x - 2y + 2z = -10

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

\newline

Multiply and Compare Equations: First, let's multiply the second equation by $2$ to see if it matches any of the other equations. $\newline$ $2(-x + y - z) = 2(5)$ $\newline$ $-2x + 2y - 2z = 10$
Analyze Equations: Now, we compare this new equation with the first and the third equations. $\newline$ The first equation is $-2x + 2y - 2z = 10$ , which is the same as the new equation we got from doubling the second one. $\newline$ The third equation is $2x - 2y + 2z = -10$ , which is the exact opposite of the first equation.
Determine Solution Type: Since the first and the third equations are opposites, adding them will result in $0 = 0$ , which is always true. $\newline$ And since the second equation is a multiple of the first, it's not independent. $\newline$ This means the system has infinitely many solutions because the equations are dependent.