To find the point of intersection for the planes given by the equations:
[tex]\[
\begin{array}{c}
7x - 2y + z = 15 \\
x + y - 3z = 4 \\
2x - y + 5z = 2
\end{array}
\][/tex]
we need to solve this system of linear equations. Our goal is to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] that satisfy all three equations simultaneously.
First, let’s summarize the system of equations:
1. [tex]\(7x - 2y + z = 15\)[/tex]
2. [tex]\(x + y - 3z = 4\)[/tex]
3. [tex]\(2x - y + 5z = 2\)[/tex]
After solving this system, we find the solution to be:
[tex]\[
\begin{cases}
x = \frac{76}{33} \\
y = \frac{1}{3} \\
z = -\frac{5}{11}
\end{cases}
\][/tex]
Therefore, the point of intersection of the three planes is [tex]\(\left( \frac{76}{33}, \frac{1}{3}, -\frac{5}{11} \right)\)[/tex].
Let's break it down for each variable:
To find [tex]\( x \)[/tex]:
- The value of [tex]\( x \)[/tex] is [tex]\(\frac{76}{33}\)[/tex].
To find [tex]\( y \)[/tex]:
- The value of [tex]\( y \)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
To find [tex]\( z \)[/tex]:
- The value of [tex]\( z \)[/tex] is [tex]\(-\frac{5}{11}\)[/tex].
So, the solution (point of intersection) is written as:
[tex]\[
\left( \frac{76}{33}, \frac{1}{3}, -\frac{5}{11} \right)
\][/tex]
This point lies on all three planes, meaning it satisfies all three equations given in the system.
