To show that the two lines intersect, we need to find a point (values for \(X\), \(y\), and \(z\)) that satisfies both sets of parametric equations. Let's solve the system of equations formed by equating the corresponding components of the two parametric equations.
For the first set of parametric equations:
\[ \frac{X - 5}{4} = \frac{y - 7}{-4} = \frac{z - 3}{-5} \]
Let's denote \(t\) as the common parameter for the two parametric equations. We have:
\[ X - 5 = 4t \]
\[ y - 7 = -4t \]
\[ z - 3 = -5t \]
Now, for the second set of parametric equations:
\[ \frac{X - 8}{4} = \frac{y - 4}{-4} = \frac{z - 5}{4} \]
Denote \(s\) as the parameter for the second set. We have:
\[ X - 8 = 4s \]
\[ y - 4 = -4s \]
\[ z - 5 = 4s \]
Now, we can set up a system of equations to find the values of \(X\), \(y\), and \(z\) that satisfy both sets of equations:
\[ X - 5 = 4t \]
\[ y - 7 = -4t \]
\[ z - 3 = -5t \]
\[ X - 8 = 4s \]
\[ y - 4 = -4s \]
\[ z - 5 = 4s \]
We can solve this system of equations to find the values of \(t\) and \(s\) and then substitute them back into one of the parametric equations to find \(X\), \(y\), and \(z\). If such values exist, then the lines intersect at that point.
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