The system of linear equations you have provided is:

\[
4x - y = 10
\]
\[
-2x + 5y = 4
\]

This system can be solved using matrices. Let's rewrite the system in matrix form:

\[
\begin{pmatrix}
4 & -1 \\
-2 & 5
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
10 \\
4
\end{pmatrix}
\]

### Step 1: Matrix Representation
The matrix equation is written as:

\[
A \mathbf{x} = \mathbf{b}
\]
Where:
- \(A\) is the coefficient matrix:
  \[
  A = \begin{pmatrix}
  4 & -1 \\
  -2 & 5
  \end{pmatrix}
  \]
- \(\mathbf{x}\) is the variable matrix:
  \[
  \mathbf{x} = \begin{pmatrix}
  x \\
  y
  \end{pmatrix}
  \]
- \(\mathbf{b}\) is the constant matrix:
  \[
  \mathbf{b} = \begin{pmatrix}
  10 \\
  4
  \end{pmatrix}
  \]

### Step 2: Find the inverse of matrix \(A\)
To solve for \(\mathbf{x}\), we use the formula:

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

Let's calculate the inverse of matrix \(A\). The inverse of a 2x2 matrix is given by:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\]

For the matrix \(A = \begin{pmatrix} 4 & -1 \\ -2 & 5 \end{pmatrix}\), the determinant \(\text{det}(A)\) is:

\[
\text{det}(A) = (4)(5) - (-1)(-2) = 20 - 2 = 18
\]

So, the inverse is:

\[
A^{-1} = \frac{1}{18} \begin{pmatrix} 5 & 1 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} \frac{5}{18} & \frac{1}{18} \\ \frac{2}{18} & \frac{4}{18} \end{pmatrix}
\]

### Step 3: Multiply \(A^{-1}\) by \(\mathbf{b}\)
Now multiply the inverse matrix by the constant matrix:

\[
A^{-1} \mathbf{b} = \begin{pmatrix} \frac{5}{18} & \frac{1}{18} \\ \frac{2}{18} & \frac{4}{18} \end{pmatrix} \begin{pmatrix} 10 \\ 4 \end{pmatrix}
\]

Carrying out the multiplication:

\[
\mathbf{x} = \begin{pmatrix} \frac{5}{18}(10) + \frac{1}{18}(4) \\ \frac{2}{18}(10) + \frac{4}{18}(4) \end{pmatrix}
= \begin{pmatrix} \frac{50}{18} + \frac{4}{18} \\ \frac{20}{18} + \frac{16}{18} \end{pmatrix}
= \begin{pmatrix} \frac{54}{18} \\ \frac{36}{18} \end{pmatrix}
= \begin{pmatrix} 3 \\ 2 \end{pmatrix}
\]

### Final Solution
The solution is:
\[
x = 3, \quad y = 2
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you calculate the determinant of a 3x3 matrix?
2. What is the geometric interpretation of solving a system of equations using matrices?
3. How can matrix multiplication be applied in real-world scenarios?
4. How does the Cramer's rule differ from using matrix inverses to solve systems?
5. Can all systems of equations be solved using matrix inverses?

### Tip:
When finding the inverse of a 2x2 matrix, always check that the determinant is non-zero; otherwise, the matrix has no inverse, and the system may not have a unique solution.

4x - y = 10
-2x + 5y = 4
solve using matrix

Learn how to solve the system of linear equations 4x - y = 10 and -2x + 5y = 4 using matrix methods. This problem involves key concepts in linear algebra, including matrix inversion and determinant calculations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation