Sure! Let’s solve the given system of linear equations step-by-step.
The system of equations is:
[tex]\[
\begin{cases}
9x + 4y - 10z = 6 \\
6x - 8y + 5z = -1 \\
12x + 12y - 15z = 10
\end{cases}
\][/tex]
### Step 1: Write the system in matrix form
We can express the system of equations in matrix form [tex]\( AX = B \)[/tex], where [tex]\( A \)[/tex] is the coefficient matrix, [tex]\( X \)[/tex] is the column vector of variables, and [tex]\( B \)[/tex] is the constant vector.
[tex]\[
A = \begin{pmatrix}
9 & 4 & -10 \\
6 & -8 & 5 \\
12 & 12 & -15
\end{pmatrix},
\quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix},
\quad
B = \begin{pmatrix}
6 \\
-1 \\
10
\end{pmatrix}
\][/tex]
### Step 2: Solve the system using matrix methods
To solve the system, we need to find [tex]\( X \)[/tex] such that [tex]\( AX = B \)[/tex]. This can be done using a variety of methods such as Gaussian elimination, matrix inversion, or other numerical methods.
### Step 3: Interpret the solution
The solution to the system of equations has been found to be:
[tex]\[
X = \begin{pmatrix}
0.333 \\
0.250 \\
-0.200
\end{pmatrix}
\][/tex]
### Step 4: Assign the values to the variables
So, the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] are:
[tex]\[
x = 0.333, \quad y = 0.250, \quad z = -0.200
\][/tex]
These values satisfy all three equations:
1. Checking the first equation [tex]\(9x + 4y - 10z = 6\)[/tex]:
[tex]\[
9(0.333) + 4(0.250) - 10(-0.200) = 2.997 + 1.000 + 2.000 = 6
\][/tex]
2. Checking the second equation [tex]\(6x - 8y + 5z = -1\)[/tex]:
[tex]\[
6(0.333) - 8(0.250) + 5(-0.200) = 1.998 - 2.000 - 1.000 = -1
\][/tex]
3. Checking the third equation [tex]\(12x + 12y - 15z = 10\)[/tex]:
[tex]\[
12(0.333) + 12(0.250) - 15(-0.200) = 3.996 + 3.000 + 3.000 = 10
\][/tex]
### Conclusion
Hence, the solution to the system of equations is:
[tex]\[
(x, y, z) = \left(\frac{1}{3}, \frac{1}{4}, -\frac{1}{5}\right) \approx (0.333, 0.250, -0.200)
\][/tex]
These values satisfy all the given equations, confirming the solution.
