Hey,
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We need to solve the system of equations simultaneously. Using the elimination method to reduce the system step by step and solve for $a$ awould work here.
The solution of the system in terms of \(a\) is given by:
\[x = \frac{20 - 18a}{5a - 18}\]
\[y = \frac{21a - 50}{5a - 18}\]
\[z = \frac{32}{5a - 18}\]
For the system to be consistent, the denominators of $x$, $y$, and $z$ must not be zero, as this would make the variables undefined. Therefore, the condition for the system to be consistent is that $5a - 18 \neq 0$.
Solving this inequality for $a$ gives:
$5a - 18 \neq 0$
$5a \neq 18$
$a \neq \frac{18}{5}$
So, the system of equations is consistent for all values of $a$ except $a = \frac{18}{5}$.
The system of equations represents three planes in 3-dimensional space. The consistency of the system implies that these planes intersect at a point, line, or coincide with each other (infinitely many solutions), depending on the values of $a$.
- For $a \neq \frac{18}{5}$, the planes intersect at a unique point in 3D space, indicating a single solution for $x$, $y$, and $z$. This means the planes are not parallel to each other and do not coincide.
- For $a = \frac{18}{5}$, the denominator becomes zero, suggesting that the equations might represent parallel planes (no intersection) or the same plane infinitely many times (infinite solutions). However, since this value makes the solution undefined, it suggests a condition where the system loses consistency, possibly due to the planes being parallel or the equations becoming dependent in a way that doesn't allow for a unique intersection point.
Thus, the value of $a$ controls the spatial relationship between these three planes, affecting whether they intersect and how.
Kav10
