How many solutions does the system of equations below have? $\newline$ $-x + 3y - z = 17$ $\newline$ $3x - y - z = -20$ $\newline$ $x - 3y + z = -17$ $\newline$ 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 = 17

\newline

3x - y - z = -20

\newline

x - 3y + z = -17

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Combine Equations: Add the first and third equations to see if they cancel each other out. $\newline$ $(-x + 3y - z) + (x - 3y + z) = 17 + (-17)$
Simplify Result: Simplify the equation. $\newline$ $0 = 0$
Check Second Equation: Since the first and third equations add up to a true statement, they might be multiples of each other, or the system has infinitely many solutions. Let's check the second equation to see if it's also a multiple of the first.
Multiply First Equation: Multiply the first equation by $3$ and compare it to the second equation. $\newline$ $3(-x + 3y - z) = 3(17)$ $\newline$ $3x - y - z = -20$
Compare to Second Equation: Simplify the multiplied equation. $\newline$ $-3x + 9y - 3z = 51$
Determine Solutions: Compare the simplified equation to the second equation. $\newline$ $-3x + 9y - 3z \neq 3x - y - z$
Subtract Equations: Since the equations are not multiples of each other, the system does not have infinitely many solutions. We need to check if there is one solution or no solution.
Simplify Result: Subtract the third equation from the first equation. $\newline$ $(-x + 3y - z) - (x - 3y + z) = 17 - (-17)$
Divide Equation: Simplify the equation. $\newline$ $-2x + 6y - 2z = 34$
Identify Same Line: Divide the equation by $-2$ to simplify further. $x - 3y + z = -17$
Check Intersection: This is the same as the third equation, which means the first and third equations are actually the same line. Now we need to check if the second equation intersects this line.
Find Constant: If the second equation can be made to look like the first and third by multiplying by a constant, then it's the same line and we have infinitely many solutions. If not, it's a different line and we have one solution.
Different Line: Try to find a constant to multiply the first equation by to get the second equation. $\newline$ There is no such constant because the signs of the $x$ terms are different.
Different Line: Try to find a constant to multiply the first equation by to get the second equation. $\newline$ There is no such constant because the signs of the $x$ terms are different.Since the second equation cannot be made to look like the first and third, it represents a different line. Therefore, the system of equations has one point of intersection.