https://file.aiquickdraw.com/mathbot/file/1728900489378ww78n6af.jpg

The system of linear equations provided in the image is:

\[
\begin{aligned}
1. & \quad x - 4y + 5z = -1 \\
2. & \quad 2x - y + 3z = 1 \\
3. & \quad 3x + 2y + z = 3
\end{aligned}
\]

To determine how many solutions this system has, we need to check the determinant of the coefficient matrix. If the determinant is non-zero, the system has a **unique solution**. If the determinant is zero, we check for consistency (to see if there are either **no solutions** or **infinite solutions**).

Let's extract the coefficient matrix:

\[
A = \begin{pmatrix}
1 & -4 & 5 \\
2 & -1 & 3 \\
3 & 2 & 1
\end{pmatrix}
\]

Now, calculate the determinant of this matrix \(A\).

The determinant is given by:

\[
\text{det}(A) = 1 \cdot \left( (-1)(1) - 3(2) \right) - (-4) \cdot \left( 2(1) - 3(3) \right) + 5 \cdot \left( 2(-1) - (-1)(3) \right)
\]

\[
\text{det}(A) = 1 \cdot \left( -1 - 6 \right) + 4 \cdot \left( 2 - 9 \right) + 5 \cdot \left( -2 + 3 \right)
\]

\[
\text{det}(A) = 1 \cdot (-7) + 4 \cdot (-7) + 5 \cdot (1)
\]

\[
\text{det}(A) = -7 - 28 + 5 = -30
\]

Since the determinant of the matrix is \(-30 \neq 0\), the system has a **unique solution**.

Thus, the correct answer is **Option A** (Unique solution).

---

Would you like more details or have any other questions?

Here are five questions that expand on this topic:

1. What methods can be used to solve a system of linear equations?
2. How does the determinant of a matrix relate to the solvability of a system?
3. Can a system of linear equations with a zero determinant ever have a solution?
4. How would you solve a system of equations with infinite solutions?
5. What is the significance of consistency in a system of linear equations?

**Tip:** Always check the determinant first when analyzing a system of equations; it quickly reveals the potential type of solutions!

Given system of linear equations: x - 4y + 5z = -1, 2x - y + 3z = 1, 3x + 2y + z = 3. Does this system have a unique solution, no solution, infinite solutions, or n-r solutions?

Analyze a system of linear equations and determine if it has a unique solution using the determinant of a 3x3 matrix. This problem is ideal for students in high school or undergraduate-level linear algebra courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation