solve the system using matrix        -2x-3y=-26       3x+4y=36

We can solve the system of equations using matrices.

The system is:

\[
\begin{aligned}
-2x - 3y &= -26 \\
3x + 4y &= 36
\end{aligned}
\]

We can express this system in matrix form as:

\[
\begin{bmatrix}
-2 & -3 \\
3 & 4
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
-26 \\
36
\end{bmatrix}
\]

This is in the form \( A \cdot X = B \), where:

\[
A = \begin{bmatrix} -2 & -3 \\ 3 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} -26 \\ 36 \end{bmatrix}
\]

### Step 1: Find the inverse of matrix \( A \).
The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is:

\[
A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]

For our matrix \( A = \begin{bmatrix} -2 & -3 \\ 3 & 4 \end{bmatrix} \), we calculate the determinant:

\[
\text{Det}(A) = (-2)(4) - (-3)(3) = -8 + 9 = 1
\]

Thus, the inverse of \( A \) is:

\[
A^{-1} = \frac{1}{1} \begin{bmatrix} 4 & 3 \\ -3 & -2 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ -3 & -2 \end{bmatrix}
\]

### Step 2: Multiply the inverse of \( A \) by \( B \) to find \( X \).
Now, we multiply \( A^{-1} \) by \( B \):

\[
X = A^{-1} \cdot B = \begin{bmatrix} 4 & 3 \\ -3 & -2 \end{bmatrix} \begin{bmatrix} -26 \\ 36 \end{bmatrix}
\]

Performing the multiplication:

\[
\begin{aligned}
x &= (4)(-26) + (3)(36) = -104 + 108 = 4 \\
y &= (-3)(-26) + (-2)(36) = 78 - 72 = 6
\end{aligned}
\]

### Final Answer:
The solution to the system is:

\[
x = 4, \quad y = 6
\]

Would you like further explanation or have any questions?

### Related Questions:
1. How do you solve a system of three equations using matrices?
2. How do you calculate the determinant of a 3x3 matrix?
3. What happens if the determinant of the matrix is zero?
4. How do you perform matrix multiplication step by step?
5. How would the solution change if the system had no solution or infinitely many solutions?

**Tip:** Always check your final solution by substituting the values of \( x \) and \( y \) back into the original equations to verify the correctness.

Solve the system using matrix: -2x - 3y = -26, 3x + 4y = 36

Learn how to solve the system of linear equations -2x - 3y = -26 and 3x + 4y = 36 using matrices. The solution involves finding the inverse of the coefficient matrix and performing matrix multiplication. This step-by-step guide is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation