Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
−2x1	 − 	3x2	 = 	−15
6x1	 + 	x2	 = 	−27


The given system of linear equations is:

\[
\begin{aligned}
-2x_1 - 3x_2 &= -15 \\
6x_1 + x_2 &= -27
\end{aligned}
\]

We can represent this system in matrix form as **Ax = b**, where:

\[
A = \begin{pmatrix} -2 & -3 \\ 6 & 1 \end{pmatrix}, \quad
x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \quad
b = \begin{pmatrix} -15 \\ -27 \end{pmatrix}
\]

Thus, the matrix equation becomes:

\[
\begin{pmatrix} -2 & -3 \\ 6 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} -15 \\ -27 \end{pmatrix}
\]

To solve for **x**, we can use the inverse of matrix **A** (denoted \( A^{-1} \)) to find:

\[
x = A^{-1}b
\]

Now let's calculate \( A^{-1} \) and solve for \( x_1 \) and \( x_2 \).The solution to the system of linear equations is:

\[
x_1 = -6, \quad x_2 = 9
\]

Thus, the solution vector \( x \) is:

\[
x = \begin{pmatrix} -6 \\ 9 \end{pmatrix}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the inverse of a matrix manually?
2. What are the conditions for a matrix to be invertible?
3. Can every system of linear equations be solved using matrix methods?
4. How do matrix determinants relate to solving linear systems?
5. What methods can be used to solve a matrix equation if \( A \) is not invertible?

**Tip:** If a matrix is singular (non-invertible), you can use other methods like row reduction or pseudo-inverses to solve the system.

Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
−2x1 − 3x2 = −15
6x1 + x2 = −27

Solve the system of linear equations −2x1 − 3x2 = −15 and 6x1 + x2 = −27 by representing it as a matrix equation Ax = b. Learn how to find the inverse of a matrix and use it to solve for the variables. Suitable for students in Grades 10-12.

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