How many solutions does the system of equations below have? x + 3y - z = -5 -x - 3y + z = 5 2x + 3y + z = 17 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Add Equations: Add the first and second equations to eliminate x, y, and z. (x + 3y - z) + (-x - 3y + z) = -5 + 5 0 = 0Check Dependence: Since 0 = 0 is a true statement, it means the first two equations are dependent, and they represent the same plane.Check Consistency: Now, check if the third equation is consistent with the first two by adding the first equation to the third. (x + 3y - z) + (2x + 3y + z) = -5 + 17 3x + 6y = 12Simplify Equation: Divide the entire equation by 3 to simplify. x + 2y = 4Different Plane: This new equation is not a multiple of the first equation, which means the third equation represents a different plane that intersects the plane represented by the first two equations at a line.Infinite Solutions: Since all three planes intersect at a line, the system has infinitely many solutions.

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How many solutions does the system of equations below have? $\newline$ $x + 3y - z = -5$ $\newline$ $-x - 3y + z = 5$ $\newline$ $2x + 3y + z = 17$ $\newline$ 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

x + 3y - z = -5

\newline

-x - 3y + z = 5

\newline

2x + 3y + z = 17

\newline

Choices:

\newline

(A) no solution

\newline

(B) one solution

\newline

Add Equations: Add the first and second equations to eliminate $x$ , $y$ , and $z$ . $\newline$ $(x + 3y - z) + (-x - 3y + z) = -5 + 5$ $\newline$ $0 = 0$
Check Dependence: Since $0 = 0$ is a true statement, it means the first two equations are dependent, and they represent the same plane.
Check Consistency: Now, check if the third equation is consistent with the first two by adding the first equation to the third. $\newline$ $(x + 3y - z) + (2x + 3y + z) = -5 + 17$ $\newline$ $3x + 6y = 12$
Simplify Equation: Divide the entire equation by $3$ to simplify. $\newline$ $x + 2y = 4$
Different Plane: This new equation is not a multiple of the first equation, which means the third equation represents a different plane that intersects the plane represented by the first two equations at a line.
Infinite Solutions: Since all three planes intersect at a line, the system has infinitely many solutions.