How many solutions does the system of equations below have? 2x - 2y + 2z = -10 -2x + y - 3z = 6 -x - y - 2z = -6 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Combine equations to eliminate x: First, let's add the first and third equations to eliminate x. (2x - 2y + 2z) + (-x - y - 2z) = -10 + (-6) x - 3y = -16Combine equations to eliminate x again: Now, let's add the second and third equations to eliminate x again. (-2x + y - 3z) + (-x - y - 2z) = 6 + (-6) -3x - 5z = 0Solve for x: We can solve the second equation for x. -3x = 5z x = -5z/3Substitute x into first equation: Substitute x = -5z/3 into the first simplified equation. (-5z/3) - 3y = -16Multiply to remove fraction: Multiply everything by 3 to get rid of the fraction. -5z - 9y = -48Two equations with two variables: Now we have two equations with two variables: 1) x - 3y = -16 2) -5z - 9y = -48 We can't solve for a unique solution without a third equation involving z and y.

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How many solutions does the system of equations below have? $\newline$ $2x - 2y + 2z = -10$ $\newline$ $-2x + y - 3z = 6$ $\newline$ $-x - y - 2z = -6$ $\newline$ 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

2x - 2y + 2z = -10

\newline

-2x + y - 3z = 6

\newline

-x - y - 2z = -6

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Combine equations to eliminate x: First, let's add the first and third equations to eliminate x. $\newline$ $(2x - 2y + 2z) + (-x - y - 2z) = -10 + (-6)$ $\newline$ $x - 3y = -16$
Combine equations to eliminate x again: Now, let's add the second and third equations to eliminate x again. $\newline$ $(-2x + y - 3z) + (-x - y - 2z) = 6 + (-6)$ $\newline$ $-3x - 5z = 0$
Solve for x: We can solve the second equation for $x$ . $-3x = 5z$ $x = -\frac{5z}{3}$
Substitute $x$ into first equation: Substitute $x = -\frac{5z}{3}$ into the first simplified equation. $\newline$ $\left(-\frac{5z}{3}\right) - 3y = -16$
Multiply to remove fraction: Multiply everything by $3$ to get rid of the fraction. $\newline$ $-5z - 9y = -48$
Two equations with two variables: Now we have two equations with two variables: $\newline$ $1$ ) $x - 3y = -16$ $\newline$ $2$ ) $-5z - 9y = -48$ $\newline$ We can't solve for a unique solution without a third equation involving $z$ and $y$ .