How many solutions does the system of equations below have? $\newline$ $x + 3y + 2z = 5$ $\newline$ $x + 2y - 2z = 16$ $\newline$ $-2x + 2y - z = -11$ $\newline$ Choices: $\newline$ $\text{[A]no solution}$ $\newline$ $\text{[B]one solution}$ $\newline$ $\text{[C]infinitely many solutions}$

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Algebra 2

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 + 2z = 5

\newline

x + 2y - 2z = 16

\newline

-2x + 2y - z = -11

\newline

Choices:

\newline

\text{[A]no solution}

\newline

\text{[B]one solution}

\newline

\text{[C]infinitely many solutions}

Write Equations: Write down the system of equations to analyze the structure. $\newline$ $x + 3y + 2z = 5$ $\newline$ $x + 2y - 2z = 16$ $\newline$ $-2x + 2y - z = -11$
Eliminate x: Subtract the second equation from the first to eliminate x and find an equation in terms of y and z. $\newline$ $(1) - (2): (x + 3y + 2z) - (x + 2y - 2z) = 5 - 16$ $\newline$ This simplifies to $y + 4z = -11$
Simplify Coefficients: Multiply the third equation by $0.5$ to simplify the coefficients. $\newline$ $0.5 \times (-2x + 2y - z) = 0.5 \times (-11)$ $\newline$ This simplifies to $-x + y - 0.5z = -5.5$
Eliminate x Again: Add the simplified third equation to the second equation to eliminate x and find another equation in terms of y and z. $\newline$ $(2) + (3): (x + 2y - 2z) + (-x + y - 0.5z) = 16 - 5.5$ $\newline$ This simplifies to $3y - 2.5z = 10.5$
Solve for $y$ and $z$ : We now have two equations with two variables: $\newline$ $y + 4z = -11$ $\newline$ $3y - 2.5z = 10.5$ $\newline$ We can use these two equations to solve for $y$ and $z$ .
Align y Terms: Multiply the first of these two equations by $3$ to align the $y$ terms: $\newline$ $3(y + 4z) = 3(-11)$ $\newline$ This simplifies to $3y + 12z = -33$
Eliminate y: Subtract the new equation from the second equation to eliminate y: $\newline$ $(3y + 12z) - (3y - 2.5z) = -33 - 10.5$ $\newline$ This simplifies to $14.5z = -43.5$
Solve for z: Solve for z: $\newline$ $14.5z = -43.5$ $\newline$ $z = -43.5 / 14.5$ $\newline$ $z = -3$
Solve for $y$ : Substitute $z = -3$ into one of the two-variable equations to solve for $y$ : $y + 4(-3) = -11$ $y - 12 = -11$ $y = -11 + 12$ $y = 1$
Substitute for x: Substitute $y = 1$ and $z = -3$ into one of the original equations to solve for x: $\newline$ $x + 3(1) + 2(-3) = 5$ $\newline$ $x + 3 - 6 = 5$ $\newline$ $x - 3 = 5$ $\newline$ $x = 5 + 3$ $\newline$ $x = 8$
Check Solution: Check the solution $(x = 8, y = 1, z = -3)$ in all three original equations to ensure it satisfies all of them. $\newline$ First equation: $8 + 3(1) + 2(-3) = 5$ $\newline$ Second equation: $8 + 2(1) - 2(-3) = 16$ $\newline$ Third equation: $-2(8) + 2(1) - (-3) = -11$
Check Solution: Check the solution $(x = 8, y = 1, z = -3)$ in all three original equations to ensure it satisfies all of them. $\newline$ First equation: $8 + 3(1) + 2(-3) = 5$ $\newline$ Second equation: $8 + 2(1) - 2(-3) = 16$ $\newline$ Third equation: $-2(8) + 2(1) - (-3) = -11$ Verify the solutions in the original equations: $\newline$ First equation: $8 + 3 + (-6) = 5$ , which is true. $\newline$ Second equation: $8 + 2 + 6 = 16$ , which is true. $\newline$ Third equation: $-16 + 2 + 3 = -11$ , which is true. $\newline$ Since the solution satisfies all three equations, the system has one solution.