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# Systems of Equations

Solving 3 x 3 Systems of Linear Equations, Row Operations, Unique/No/Infinite Solutions…

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##### Question 1

no calculator

easy

[Maximum mark: 6]

The system of equations given below represents three planes in space.

\begin{aligned} x + 5z &= 2 \\[6pt] -2x + y - 6z &= -1 \\[6pt] 2y + 8z &= 6\end{aligned}
1. Show that this system of equations has an infinite number of solutions. 

2. Find the parametric equations of the line of intersection of the three planes. 

easy

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##### Question 2

no calculator

medium

[Maximum mark: 5]

The system of equations given below represents three planes in space.

\begin{aligned} x + 2y - 2z &= 4 \\[6pt] 3x + 5y - 4z &= 9 \hspace{3em} [\text{where a,b \in \mathbb{R}}] \\[6pt] \hspace{8em} 4x + 6y + az &= b\end{aligned}

Find the set of values of $a$ and $b$ such that the three planes have no points of
intersection.

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##### Question 3

calculator

medium

[Maximum mark: 6]

The system of equations given below represents three planes in space.

\begin{aligned} x - 3y + 2z &= 5 \\[6pt] 3x + 5y + az &= b \hspace{3em} [\text{where a,b \in \mathbb{R}}] \\[6pt] \hspace{8em} 4x + 2y - 3z &= 7\end{aligned}

Find the set of values of $a$ and $b$ such that the three planes have exactly one intersection point.

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##### Question 4

no calculator

medium

[Maximum mark: 9]

Consider the following system of equations:

\begin{aligned} x - 3z &= -2 \\[6pt] -3x + y + 6z &= 3 \\[6pt] 2x - 2y + (a-4)z &= b-3\end{aligned}

where $a,b \in \mathbb{R}$.

1. Find conditions on $a$ and $b$ for which

1. the system has no solutions;

2. the system has only one solution;

3. the system has an infinite number of solutions. 

2. In the case where the number of solutions is infinite, find the general
solution of the system of equations in Cartesian form. 

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##### Question 5

no calculator

medium

[Maximum mark: 9]

Consider the following system of equations:

\begin{aligned} x + y + 4z &= 1 \\[6pt] 3x + 2y + 16z &= 5 \\[6pt] 4x + 2y + (a-1)z &= b-4\end{aligned}

where $a,b \in \mathbb{R}$.

1. Find conditions on $a$ and $b$ for which

1. the system has no solutions;

2. the system has only one solution;

3. the system has an infinite number of solutions. 

2. In the case where the number of solutions is infinite, find the general
solution of the system of equations in Cartesian form. 

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##### Question 6

no calculator

medium

[Maximum mark: 8]

Consider the following system of equations:

\begin{aligned} x + y + z &= -1 \\[6pt] 4x + 2y + z &= 3 \\[6pt] 9x + 3y &= p\end{aligned}

where $p \in \mathbb{R}$.

1. Show that this system does not have a unique solution for any value of $p$. 

1. Determine the value of $p$ for which the system is consistent.

2. For this value of $p$, find the general solution of the system. 

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