Solve
-5=7a-4b-c
1.5=-4a+15b-5c
-10=-a-5b+6c

To solve the system of linear equations:

\[
\begin{aligned}
&-5 = 7a - 4b - c \quad \text{(Equation 1)} \\
&1.5 = -4a + 15b - 5c \quad \text{(Equation 2)} \\
&-10 = -a - 5b + 6c \quad \text{(Equation 3)}
\end{aligned}
\]

we can approach it using various methods, such as substitution, elimination, or matrix operations. I'll use the matrix method to find the solution.

### Step 1: Write the system as a matrix equation
The system can be expressed in matrix form as:
\[
\begin{bmatrix} 7 & -4 & -1 \\ -4 & 15 & -5 \\ -1 & -5 & 6 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} -5 \\ 1.5 \\ -10 \end{bmatrix}
\]

### Step 2: Solve for \( a \), \( b \), and \( c \)
To solve the system, we can find the inverse of the coefficient matrix (if it exists) and then multiply it by the constant matrix. Let me proceed with the calculations.

I'll perform the computations now.The solution to the system of equations is:

\[
a = -2.0773, \quad b = -1.5576, \quad c = -3.3109
\]

These are approximate values, rounded to four decimal places.

Would you like more details on the solving process, or have any questions about this solution?

### Further Exploration:
1. How can we verify that this solution satisfies the original equations?
2. What are some alternative methods to solve a system of linear equations?
3. How would the solution change if one of the equations was altered?
4. What is the importance of checking if the determinant of the coefficient matrix is zero?
5. Can this system be solved graphically, and what would that entail?

**Tip:** Always check your solution by substituting the values back into the original equations. This ensures the accuracy of the result.

Solve the system of equations:
-5 = 7a - 4b - c
1.5 = -4a + 15b - 5c
-10 = -a - 5b + 6c

Learn how to solve a system of linear equations using matrix inversion. This example demonstrates step-by-step calculations for the equations: -5 = 7a - 4b - c, 1.5 = -4a + 15b - 5c, and -10 = -a - 5b + 6c. Explore key concepts in linear algebra suitable for Grades 11-12.

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