Math Problem Statement
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?
Solution
The system of linear equations provided in the image is:
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!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Determinants
Formulas
Determinant of a 3x3 matrix
Cramer's Rule
Theorems
Solvability of systems based on determinants
Uniqueness Theorem
Suitable Grade Level
Grades 10-12 or Undergraduate
Related Recommendation
Proving the Uniqueness of Solutions for a System with Determinant 3
Solving Homogeneous System of Equations with Determinants
Determine Unique Solutions for a System of Equations with Parameter m
Understanding System Compatibility Through Determinants
Find Values of c for a System of Equations: Unique, Infinite, or No Solutions