To find the point of intersection for the planes described by the following equations, we need to solve the system of linear equations:
[tex]\[
\begin{aligned}
&7x - 2y + z = 15, \\
&x + y - 3z = 4, \\
&2x - y + 5z = 2.
\end{aligned}
\][/tex]
Step by step, our goal is to determine the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
1. Setting up the system of equations:
[tex]\[
\begin{aligned}
7x - 2y + z &= 15, \\
x + y - 3z &= 4, \\
2x - y + 5z &= 2.
\end{aligned}
\][/tex]
2. Express in matrix form:
We can write the system as a matrix equation [tex]\(AX = B\)[/tex], where:
[tex]\[
A = \begin{pmatrix}
7 & -2 & 1 \\
1 & 1 & -3 \\
2 & -1 & 5
\end{pmatrix}, \quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad
B = \begin{pmatrix}
15 \\
4 \\
2
\end{pmatrix}.
\][/tex]
3. Solving the system:
To find [tex]\(X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)[/tex], we solve the matrix equation [tex]\(AX = B\)[/tex]. The solution involves finding the inverse of matrix [tex]\(A\)[/tex] and then multiplying it by [tex]\(B\)[/tex], or using other efficient algorithms in linear algebra.
4. Conclusion:
Solving the system, we find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[
\begin{aligned}
x &\approx 2.303, \\
y &\approx 0.333, \\
z &\approx -0.455.
\end{aligned}
\][/tex]
So, the point of intersection for the given planes is approximately:
[tex]\[
(x, y, z) \approx (2.303, 0.333, -0.455).
\][/tex]
