Question

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

Short answer

For one solution: \( k \neq 2 \); no solution: \( k = 2 \); infinite: never.

Step-by-step solution

Set up the system of equations

We are given the system of equations: 1. \( x_1 + 2x_2 = 1 \) 2. \( x_1 + kx_2 = 2 \).

Analyze the equations to find relationships

We notice that both equations have the same left-hand term \( x_1 \). Subtract the first equation from the second: \((x_1 + kx_2) - (x_1 + 2x_2) = 2 - 1\). Simplifying gives \((k - 2)x_2 = 1\).

Determine condition for exactly one solution

For the system to have exactly one solution, \((k - 2)x_2 = 1\) should yield a unique \(x_2\). This occurs if \(k - 2 eq 0\). Therefore, \(k eq 2\).

Condition for no solution

For no solution, the system's equations must be inconsistent, meaning they become parallel without ever intersecting. This happens if the equations are dependent, requiring \(k - 2 = 0\) , which means \(k = 2\). However, this leads to \(x_1 + 2x_2\) equations having different constants (since 1 cannot equal 2), so the solution is impossible and inconsistent.

Condition for infinite solutions

Infinite solutions occur when the two equations are identical (essentially one equation). This isn't possible here because any identical solution requires \(1 = 2\), which cannot be true for any \(k\). Hence, for \(k = 2\), no solutions exist. Infinite solutions won't occur for any value of \(k\).

Key concepts

Unique Solution

In a system of equations, a unique solution means there is exactly one set of values for the variables that satisfies all equations simultaneously. For our given system, \(x_1 + 2x_2 = 1\) and \(x_1 + kx_2 = 2\), we want to determine when these equations meet at exactly one point.
To do this, we manipulate the equations to find a condition on \(k\). By subtracting the first equation from the second, we obtain \((k - 2)x_2 = 1\).
This equation suggests that for a unique solution to exist, the coefficient \(k - 2\) of \(x_2\) must not be zero.
If \(k - 2\) is zero, the equation becomes inconsistent. Thus, for a unique solution, \(k\) must not equal 2. Hence:

• When \(k eq 2\), the system has a unique solution because it leads to a non-zero multiple of \(x_2\), making the equations intersect at exactly one point.

Infinite Solutions

Infinite solutions occur when the system of equations describes the same line, meaning all solutions of one equation are solutions of the other. For this to happen in our system, the equations must be dependent, essentially boiling down to the same expression.
However, when we analyze the given equations, \(x_1 + 2x_2 = 1\) and \(x_1 + kx_2 = 2\), the condition \(1 = 2\) must hold true for them to be identical, which is impossible.
This indicates there are no values for \(k\) that make the equations infinitely solvable. Therefore:

• No value of \(k\) can lead to infinite solutions because for identical equations, a contradiction like \(1 = 2\) cannot occur.

No Solution

A system has no solution when the equations represent parallel lines that never intersect. For the given system, we identify the conditions leading to parallel lines.
Subtracting the first from the second, we have \((k - 2)x_2 = 1\). If \(k - 2 = 0\), it implies the lines are parallel with different constant terms, which causes an inconsistency.
Specifically, substituting \(k = 2\) results in \(x_1 + 2x_2 = 1\) and \(x_1 + 2x_2 = 2\).
These equations can't be true simultaneously because one cannot equal two.
Thus, for \(k = 2\), no solution exists:

• When \(k = 2\), the equations are inconsistent, resulting in parallel lines that do not intersect, thus no solution is possible.

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. These systems can be solved to find values for the variables that satisfy all equations.
In our current system, \(x_1 + 2x_2 = 1\) and \(x_1 + kx_2 = 2\), we are examining under what conditions these two equations have unique, infinite, or no solutions.
We analyze different scenarios based on the parameter \(k\) to discern the behavior of the system.
This involves:

• Identifying whether one solution exists, which occurs when the equations intersect at exactly one point.
• Determining if infinite solutions exist, which requires the lines to be identical.
• Checking if there are no solutions, which happens when the lines are parallel and non-intersecting.

Understanding these outcomes and relationships helps in solving and graphically interpreting systems of equations effectively.