Math Problem Statement
determine a, b and c so that the following equation has a, unique question b, infinity many solution c, no solution 1. x+y-bz=1, 2x+3y+az=3 and x+ay + 3z=2 2. x+2y-3y=a, 2x +6y -11z=b and x-2y+7z=c 3. x-2y+bz=3, ax+2z=2 and 5x+2y=2
Solution
To determine , , and for the given equations under each condition (unique solution, infinitely many solutions, and no solution), we use the principles of linear algebra. Specifically, we examine the system of equations in terms of their consistency and the rank of the coefficient matrix.
For each system of equations:
System 1
1. & \quad x + y - bz = 1 \\ 2. & \quad 2x + 3y + az = 3 \\ 3. & \quad x + ay + 3z = 2 \end{aligned}$$ 1. **Unique Solution:** - A unique solution exists if the determinant of the coefficient matrix is nonzero, i.e., $$\text{det}(A) \neq 0$$. - Coefficient matrix: $$A = \begin{bmatrix} 1 & 1 & -b \\ 2 & 3 & a \\ 1 & a & 3 \end{bmatrix}.$$ Compute $$\text{det}(A)$$ and solve for $$a$$ and $$b$$ such that $$\text{det}(A) \neq 0$$. 2. **Infinitely Many Solutions:** - Infinitely many solutions occur if $$\text{det}(A) = 0$$ and the augmented matrix has the same rank as the coefficient matrix. - Solve $$\text{det}(A) = 0$$ for $$a$$ and $$b$$. 3. **No Solution:** - No solution occurs if $$\text{det}(A) = 0$$, but the rank of the augmented matrix is greater than the rank of the coefficient matrix (inconsistent system). --- ### **System 2** $$\begin{aligned} 1. & \quad x + 2y - 3z = a \\ 2. & \quad 2x + 6y - 11z = b \\ 3. & \quad x - 2y + 7z = c \end{aligned}$$ 1. **Unique Solution:** - Coefficient matrix: $$A = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 6 & -11 \\ 1 & -2 & 7 \end{bmatrix}.$$ Compute $$\text{det}(A) \neq 0$$. 2. **Infinitely Many Solutions:** - Solve $$\text{det}(A) = 0$$ and check rank consistency. 3. **No Solution:** - Solve $$\text{det}(A) = 0$$ and verify the rank inconsistency between the coefficient matrix and the augmented matrix. --- ### **System 3** $$\begin{aligned} 1. & \quad x - 2y + bz = 3 \\ 2. & \quad ax + 2z = 2 \\ 3. & \quad 5x + 2y = 2 \end{aligned}$$ 1. **Unique Solution:** - Coefficient matrix: $$A = \begin{bmatrix} 1 & -2 & b \\ a & 0 & 2 \\ 5 & 2 & 0 \end{bmatrix}.$$ Solve $$\text{det}(A) \neq 0$$. 2. **Infinitely Many Solutions:** - Solve $$\text{det}(A) = 0$$ and check rank consistency. 3. **No Solution:** - Solve $$\text{det}(A) = 0$$ and verify the rank inconsistency between the coefficient matrix and the augmented matrix. --- Would you like me to compute these determinants and solve for $$a, b, c$$ for specific cases? Let me know! 😊 --- ### **Relative Questions** 1. What does the determinant of a matrix signify in solving systems of linear equations? 2. How do you determine the rank of a matrix, and why is it important? 3. What are augmented matrices, and how do they relate to the solution of a system of equations? 4. How can consistency of a system of equations be verified? 5. What are the geometric interpretations of unique, infinite, and no solutions in systems of equations? **Tip:** When solving linear systems, always verify if the determinant is zero before proceeding, as it determines whether the system is singular or non-singular.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Determinants
Matrix Rank
Consistency of Systems
Formulas
det(A) ≠ 0 for unique solution
det(A) = 0 for infinite or no solution
Rank consistency between coefficient and augmented matrix
Theorems
Rank-Nullity Theorem
Cramer's Rule
Inconsistent Systems Theorem
Suitable Grade Level
Grades 11-12
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