https://file.aiquickdraw.com/mathbot/file/17231172894324oe685uf.jpg

The image provided contains a matrix equation that needs to be solved. Let's go through it step by step.

### The Problem Statement
The problem appears to be:

\[
\begin{aligned}
&\text{Given:} \\
&\mathbf{A} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 3 \\ 5 & 4 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} x \\ y \end{pmatrix} \\
&\text{Solve for} \ \mathbf{C} \ \text{in the equation} \ \mathbf{A} \cdot \mathbf{C} + \mathbf{B} \cdot \mathbf{C} = \mathbf{D}
\end{aligned}
\]

where:

\[
\mathbf{D} = \begin{pmatrix} 7 \\ 8 \end{pmatrix}
\]

### Step 1: Express the Equation
Given that \(\mathbf{A}\) is the identity matrix, the equation can be rewritten as:

\[
\mathbf{C} + \mathbf{B} \cdot \mathbf{C} = \mathbf{D}
\]

### Step 2: Substitute Matrices
Substituting \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{D}\) into the equation:

\[
\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 2 & 3 \\ 5 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 8 \end{pmatrix}
\]

### Step 3: Matrix Multiplication
First, multiply \(\mathbf{B}\) and \(\mathbf{C}\):

\[
\mathbf{B} \cdot \mathbf{C} = \begin{pmatrix} 2 & 3 \\ 5 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2x + 3y \\ 5x + 4y \end{pmatrix}
\]

### Step 4: Add the Vectors
Now, add the resulting vector to \(\mathbf{C}\):

\[
\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 2x + 3y \\ 5x + 4y \end{pmatrix} = \begin{pmatrix} 7 \\ 8 \end{pmatrix}
\]

This simplifies to:

\[
\begin{pmatrix} x + 2x + 3y \\ y + 5x + 4y \end{pmatrix} = \begin{pmatrix} 7 \\ 8 \end{pmatrix}
\]

Which further simplifies to:

\[
\begin{pmatrix} 3x + 3y \\ 5x + 5y \end{pmatrix} = \begin{pmatrix} 7 \\ 8 \end{pmatrix}
\]

### Step 5: Set Up the Equations
Now, equate the components:

\[
\begin{aligned}
3x + 3y &= 7 \quad \text{(Equation 1)} \\
5x + 5y &= 8 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 6: Solve the System of Equations
We can simplify both equations by dividing by the common factor:

\[
\begin{aligned}
x + y &= \frac{7}{3} \quad \text{(Equation 1 simplified)} \\
x + y &= \frac{8}{5} \quad \text{(Equation 2 simplified)}
\end{aligned}
\]

However, these simplified equations suggest a contradiction since the same expression \(x + y\) cannot equal two different values simultaneously.

Thus, the system of equations has no solution, indicating that there's likely an inconsistency in the problem setup or the values provided.

### Conclusion:
The system as presented does not have a solution. This typically suggests that the provided equations are inconsistent. If you were expecting a solution, there might be a mistake in the problem setup or a need for further clarification on the matrices.

Would you like further details or have any questions? 

Here are some related questions that you might find helpful:
1. How do you solve systems of linear equations with matrices?
2. What is the determinant of a matrix and how does it relate to solutions?
3. Can a system of linear equations have no solution, one solution, or infinitely many solutions?
4. How do you interpret the inconsistency in a system of linear equations?
5. What are eigenvalues and eigenvectors, and how do they relate to matrices?
6. How can we use row reduction to solve matrix equations?
7. What is the significance of the identity matrix in matrix operations?
8. How do matrix equations apply to real-world problems?

**Tip:** When solving matrix equations, always check if the system is consistent by ensuring the equations do not contradict each other.

Learn how to solve a matrix equation involving matrices A and B. This problem covers matrix algebra and systems of linear equations, focusing on matrix multiplication. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation