https://r2.erweima.ai/mathbot/image/1725669293865a75igbwl.jpg

The image you uploaded presents a system of two linear equations, and the task is to complete the determinants needed to solve the system using Cramer's Rule. The equations shown in the image are:

\[
\begin{cases}
-5x + 3y = 1 \\
2x - y = 5
\end{cases}
\]

To solve the system using Cramer's Rule, we need to calculate the determinants for \(x\) and \(y\) as follows:

### Step 1: Set up the coefficient matrix

The system can be represented in matrix form as:

\[
\begin{pmatrix}
-5 & 3 \\
2 & -1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
1 \\
5
\end{pmatrix}
\]

### Step 2: Determinant of the coefficient matrix (\(D\))

\[
D = \begin{vmatrix} -5 & 3 \\ 2 & -1 \end{vmatrix}
\]

Calculating \(D\):

\[
D = (-5)(-1) - (3)(2) = 5 - 6 = -1
\]

### Step 3: Determinant for \(x\) (\(D_x\))

To find \(D_x\), replace the first column of the coefficient matrix with the constants from the right-hand side of the equations:

\[
D_x = \begin{vmatrix} 1 & 3 \\ 5 & -1 \end{vmatrix}
\]

Calculating \(D_x\):

\[
D_x = (1)(-1) - (3)(5) = -1 - 15 = -16
\]

### Step 4: Determinant for \(y\) (\(D_y\))

To find \(D_y\), replace the second column of the coefficient matrix with the constants from the right-hand side:

\[
D_y = \begin{vmatrix} -5 & 1 \\ 2 & 5 \end{vmatrix}
\]

Calculating \(D_y\):

\[
D_y = (-5)(5) - (1)(2) = -25 - 2 = -27
\]

### Step 5: Solve for \(x\) and \(y\)

Using Cramer's Rule:

\[
x = \frac{D_x}{D} = \frac{-16}{-1} = 16
\]

\[
y = \frac{D_y}{D} = \frac{-27}{-1} = 27
\]

Thus, the solution to the system of equations is:

\[
x = 16, \quad y = 27
\]

---

Let me know if you'd like further clarification or more details on any step! Here are some related questions you can explore:

1. What is Cramer's Rule, and when can it be used?
2. How do you compute the determinant of a 2x2 matrix?
3. Can Cramer's Rule be applied to systems with more than two variables?
4. What happens if the determinant of the coefficient matrix is zero?
5. How would you solve a system of equations using substitution or elimination instead of Cramer's Rule?

**Tip:** When using Cramer's Rule, always check that the determinant of the coefficient matrix is non-zero; otherwise, the system has no unique solution!

Completa los determinantes que permiten solucionar el sistema de ecuaciones -5x + 3y = 1, 2x - y = 5

Learn how to solve the system of linear equations -5x + 3y = 1 and 2x - y = 5 using Cramer's Rule. This step-by-step solution explains how to calculate the determinants and find the values of x and y. Suitable for high school students (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