Finding 'a' in a System of Linear Equations

Math Problem Statement

find the value of “a” such that the following system of linear

equations

𝒙 + 𝒚 + 𝒂𝒛 = 𝟑, 𝒙 + 𝒂𝒚 + 𝒛 = 𝟐, 𝒂𝒙 + 𝒚 − 𝒛 = 𝟏

have

i) exactly one solution

ii) No solution

iii) Infinite number of solutions and find them

put it in matrix and get triangle of zeros

To solve for the values of $a$ such that the system of equations:

x + y + az = 3 \\ x + ay + z = 2 \\ ax + y - z = 1 \end{cases}$$ has: 1. Exactly one solution 2. No solution 3. An infinite number of solutions we can put this system in matrix form and perform row operations to get it into an upper triangular (triangular of zeros) form. ### Step 1: Write the system in matrix form The augmented matrix for the system is: $$\begin{bmatrix} 1 & 1 & a & | & 3 \\ 1 & a & 1 & | & 2 \\ a & 1 & -1 & | & 1 \\ \end{bmatrix}$$ ### Step 2: Perform Gaussian elimination to get upper triangular form We'll aim to make zeros below the main diagonal by performing row operations. #### Row Operations: 1. **Subtract $$\text{Row 1}$$ from $$\text{Row 2}$$:** $$\text{Row 2} = \text{Row 2} - \text{Row 1} \Rightarrow \begin{bmatrix} 1 & 1 & a & | & 3 \\ 0 & a-1 & 1-a & | & -1 \\ a & 1 & -1 & | & 1 \end{bmatrix}$$ 2. **Subtract $$a \times \text{Row 1}$$ from $$\text{Row 3}$$:** $$\text{Row 3} = \text{Row 3} - a \times \text{Row 1} \Rightarrow \begin{bmatrix} 1 & 1 & a & | & 3 \\ 0 & a-1 & 1-a & | & -1 \\ 0 & 1-a & -1-a^2 & | & 1 - 3a \end{bmatrix}$$ 3. **Next, eliminate the second element in the third row by using $$\text{Row 2}$$:** Subtract $$\frac{1-a}{a-1} \times \text{Row 2}$$ from $$\text{Row 3}$$: (Continue until upper triangular form is reached.)

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Linear Algebra
Systems of Equations
Gaussian Elimination

Ax = b
Row reduction to echelon form

Existence and uniqueness theorem for linear systems

Grades 10-12

Determining Solutions for a System of Linear Equations Involving Parameter 'a'

System of Linear Equations: Determining Values for Unique, Many, and No Solutions

Solving a System of Linear Equations with Parameter a Using Row Operations

Gaussian Elimination: Solving a System of Linear Equations with Parameter a

Solve System of Linear Equations for No, Unique, and Infinite Solutions