https://file.aiquickdraw.com/mathbot/file/17262602859647mlqc8tc.jpg

The given equation involves matrix multiplication. Here's the system we have:

\[
\begin{bmatrix}
x + y & y \\
2x & x - y
\end{bmatrix}
\begin{bmatrix}
2 \\
-1
\end{bmatrix}
=
\begin{bmatrix}
3 \\
2
\end{bmatrix}
\]

To solve this, we perform the matrix multiplication on the left-hand side:

### First row multiplication:
\[
(x + y)(2) + (y)(-1) = 3
\]
Expanding:
\[
2(x + y) - y = 3
\]
Simplifying:
\[
2x + 2y - y = 3 \quad \Rightarrow \quad 2x + y = 3 \quad \text{(Equation 1)}
\]

### Second row multiplication:
\[
(2x)(2) + (x - y)(-1) = 2
\]
Expanding:
\[
4x - (x - y) = 2
\]
Simplifying:
\[
4x - x + y = 2 \quad \Rightarrow \quad 3x + y = 2 \quad \text{(Equation 2)}
\]

Now, we have the system of equations:
1. \( 2x + y = 3 \)
2. \( 3x + y = 2 \)

Let's solve this system of linear equations.

### Step 1: Subtract Equation 1 from Equation 2
\[
(3x + y) - (2x + y) = 2 - 3
\]
\[
3x + y - 2x - y = -1
\]
\[
x = -1
\]

### Step 2: Substitute \( x = -1 \) into Equation 1
\[
2(-1) + y = 3
\]
\[
-2 + y = 3 \quad \Rightarrow \quad y = 5
\]

Thus, the solution is:
\[
x = -1, \quad y = 5
\]

Let me know if you need further details or have any questions!

---

Here are five related questions:
1. How do you check if a system of equations is consistent?
2. What happens if the determinant of a matrix is zero?
3. Can you always solve a system of linear equations using matrix multiplication?
4. How do you interpret matrix multiplication geometrically?
5. What other methods can be used to solve systems of equations?

**Tip:** When solving matrix equations, always check if the determinant is non-zero to ensure the system has a unique solution!

Solve the system of equations resulting from the matrix multiplication: [x + y, y; 2x, x - y] * [2; -1] = [3; 2].

Learn how to solve the system of equations derived from matrix multiplication: [x + y, y; 2x, x - y] * [2; -1] = [3; 2]. This problem explores key algebraic concepts and matrix operations, suitable for high school 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