Question

State for which values of \(k\) the given system will have exactly 1 solution, infinite solutions, or no solution. $$ \begin{array}{l} x_{1}+2 x_{2}=1 \\ x_{1}+k x_{2}=1 \end{array} $$

Short answer

Infinite solutions when \( k = 2 \); one solution for \( k \neq 2 \); no solutions never occur.

Step-by-step solution

Write the System in Matrix Form

Convert the given system of equations into an augmented matrix form, which is a useful format for analyzing whether the system has a solution. The system is: \[\begin{aligned}&x_1 + 2x_2 = 1 \&x_1 + kx_2 = 1\end{aligned}\] In matrix form, this becomes: \[\begin{bmatrix} 1 & 2 \1 & k \end{bmatrix} \begin{bmatrix} x_1 \x_2 \end{bmatrix} = \begin{bmatrix} 1 \1 \end{bmatrix}\] Augmented matrix: \[\begin{bmatrix} 1 & 2 & | & 1 \1 & k & | & 1 \end{bmatrix}\]

Perform Row Operations

To simplify, perform row operations to bring the matrix closer to row-echelon form, which will make it easier to determine the conditions for solutions. Subtract the first row from the second row. The new matrix is: \[\begin{bmatrix} 1 & 2 & | & 1 \0 & k-2 & | & 0 \end{bmatrix}\] This matrix will help us decide if there are one, infinite, or no solutions based on the value of \( k \).

Determine Infinite Solutions

For the system to have infinite solutions, the second row must be a zero row, indicating that the two equations are dependent and describe the same line. Set \( k-2 = 0 \). Solving this gives \( k = 2 \). Thus, when \( k = 2 \), the system has an infinite number of solutions.

Determine No Solutions

For no solution, the second row in row-echelon form must indicate a contradiction. This occurs when the coefficient of \( x_2 \) is zero but the row itself is inconsistent. Since the system doesn't include a free variable to create an inconsistency, a contradiction cannot occur in this specific system regardless of \( k \). Thus, there are no values of \( k \) for which the system has no solutions.

Determine Exactly One Solution

The system will have exactly one solution when there is a pivot (non-zero leading coefficient) in every row of the coefficient matrix, indicating each variable can be solved uniquely. This occurs when \( k-2 eq 0 \), meaning \( k eq 2 \). Therefore, for any \( k \) except \( k = 2 \), the system has exactly one solution.

Key concepts

Augmented Matrix

An augmented matrix is a powerful tool for simplifying and solving systems of linear equations. It combines the coefficients of the variables and the constants from each equation into one neat matrix.
This helps us focus purely on the relationships between the numbers, rather than getting tangled in the variables themselves.
The given system of equations is:

• \(x_1 + 2x_2 = 1\)
• \(x_1 + kx_2 = 1\)

In augmented matrix form, the system looks like this:\[\begin{bmatrix} 1 & 2 & | & 1 \ 1 & k & | & 1 \end{bmatrix}\] This matrix neatly organizes all necessary components of a system, allowing for a streamlined approach to finding solutions.

Row Operations

Row operations are the steps taken to simplify matrices. They are akin to manipulating equations, but more systematically.
You can think of them as tools to transform the matrix into a simpler version, like row-echelon form, which is easier to interpret.There are three primary row operations we use:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting the multiples of one row from another

For this system, we subtract the first row from the second, to make our calculations clearer:\[\begin{bmatrix} 1 & 2 & | & 1 \ 0 & k-2 & | & 0 \end{bmatrix}\]These operations transform the matrix in a way that reveals important insights about the solution set.

Row-Echelon Form

The row-echelon form is a specific layout for matrices that simplifies the process of finding solutions to linear systems.
In this form, we seek to have zeros below any leading terms (non-zero coefficients) in each row.Take our matrix:\[\begin{bmatrix} 1 & 2 & | & 1 \ 0 & k-2 & | & 0 \end{bmatrix}\]We've made progress towards row-echelon form by achieving a zero in the first column of the second row. This arrangement helps uncover relationships between equations:
- Leading terms indicate pivots, marking solutions.- Zeros signify dependency or constraints, crucial for checking infinite solutions.Row-echelon form simplifies analysis, making it easier to discern exact or infinite solutions, or even contradictions.

Infinite Solutions

Infinite solutions occur when a system of equations doesn't just have one specific answer, but rather a "family" of solutions.For the system to have infinite solutions, the equations must essentially represent the same geometric line.
This scenario happens when the non-zero rows of our matrix confirm redundancy.In our case, infinite solutions occur if the second row becomes a zero row:\(k-2 = 0\)Solving this gives us \(k = 2\), signifying that both equations are identical, depicting the same line.
When this occurs, any solution of one equation will fit the other, hence, infinitely many solutions exist.

Unique Solution

A unique solution implies there's only one set of values for the variables that satisfies all the equations simultaneously.
This happens when the matrix has a definite structure after simplification through row operations.For a unique solution in our system, each equation or row in the matrix should independently contribute new information. This means avoiding scenarios that lead to zeros, which would indicate dependency:\(k eq 2\)If \(k\) is any value other than 2, the modified matrix:\[\begin{bmatrix} 1 & 2 & | & 1 \ 0 & k-2 & | & 0 \end{bmatrix}\]will have non-zero pivots in each row.
This confirms that every variable can be uniquely determined, resulting in just one unique solution.