Answer :
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]
[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]