Solve the system of equations:

[tex]\[ \left\{\begin{array}{l}9x + 4y - 10z = 6 \\ 6x - 8y + 5z = -1 \\ 12x + 12y - 15z = 10\end{array}\right. \][/tex]



Answer :

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.