Understanding System of Linear Equations and Unique Solutions

A system of linear equations consists of two or more linear equations with the same variables. When we talk about a 'solution' to such a system, we mean the values of the variables that make all the equations true at the same time. A 'unique solution' means there is exactly one specific set of values for the variables that satisfies all the equations.

Condition for a Unique Solution of Linear Equations

Consider a general system of two linear equations in two variables, \(x\) and \(y\):

• \( a_1x + b_1y = c_1 \)
• \( a_2x + b_2y = c_2 \)

This system will have a unique solution if and only if the ratio of the coefficients of \(x\) is not equal to the ratio of the coefficients of \(y\). In other words, the lines represented by the equations must intersect at a single point.

The mathematical condition for a unique solution is:

\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)

Alternatively, this condition can be expressed as \( a_1b_2 - a_2b_1 \neq 0 \).

Applying the Unique Solution Condition

The given system of equations is:

• Equation 1: \( 7x + ky = 27 \)
• Equation 2: \( kx + 7y = 19 \)

By comparing these equations to the general form \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \), we can identify the coefficients:

• From Equation 1: \( a_1 = 7 \), \( b_1 = k \)
• From Equation 2: \( a_2 = k \), \( b_2 = 7 \)

For this system to have a unique solution, the condition \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) must hold true.

Substituting the identified coefficients, we get the inequality:

\( \frac{7}{k} \neq \frac{k}{7} \)

Determining the Value of k for Unique Solution

To find the condition on \(k\) for the system to have a unique solution, we solve the inequality \( \frac{7}{k} \neq \frac{k}{7} \).

We can cross-multiply the terms (assuming \( k \neq 0 \). If \( k=0 \), the equations become \( 7x=27 \) and \( 7y=19 \), which clearly has a unique solution \( x=27/7, y=19/7 \), so \( k=0 \) is allowed.):

\( 7 \times 7 \neq k \times k \)

\( 49 \neq k^2 \)

To satisfy this inequality, \(k\) must not be the square root of 49 or the negative square root of 49.

\( k \neq \sqrt{49} \) and \( k \neq -\sqrt{49} \)

This gives us the condition:

\( k \neq 7 \) and \( k \neq -7 \)

So, for the system to have a unique solution, \(k\) must not be equal to 7 and \(k\) must not be equal to -7.

Evaluating the Options based on the Unique Solution Condition

We are asked which of the given options is correct if the system has a unique solution. This means we need to find which statement is true when the condition \( k \neq 7 \) and \( k \neq -7 \) is met.

• Option 1: \( k \neq 7 \)
  If the system has a unique solution, then based on our derivation, it is required that \( k \neq 7 \) (along with \( k \neq -7 \)). Thus, if a unique solution exists, the statement \( k \neq 7 \) is correct.
• Option 2: \( k \neq 13 \)
  If \( k = 13 \), the condition for a unique solution \( k \neq 7 \) and \( k \neq -7 \) is satisfied (\( 13 \neq 7 \) and \( 13 \neq -7 \)). This means if \( k = 13 \), the system does have a unique solution. Therefore, the statement \( k \neq 13 \) is not a necessary condition for the existence of a unique solution; the system can have a unique solution when \( k \) is 13. So this statement is not always correct when a unique solution exists.
• Option 3: \( k = 7 \)
  If \( k = 7 \), our condition for a unique solution \( k \neq 7 \) is violated. Let's check what happens when \( k=7 \). The equations become \( 7x + 7y = 27 \) and \( 7x + 7y = 19 \). Here, \( \frac{a_1}{a_2} = \frac{7}{7} = 1 \), \( \frac{b_1}{b_2} = \frac{7}{7} = 1 \), but \( \frac{c_1}{c_2} = \frac{27}{19} \). Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the system has no solution (inconsistent). Therefore, the statement \( k = 7 \) is false if the system has a unique solution.
• Option 4: \( k = 13 \)
  As discussed in Option 2, \( k = 13 \) leads to a unique solution. However, the existence of a unique solution does not imply that \( k \) *must* be 13. \( k \) could be any value other than 7 or -7. So, the statement \( k = 13 \) is not a correct condition derived from the fact that a unique solution exists.

Conclusion: The Correct Condition on k

Based on the condition for a unique solution, \( k \neq 7 \) and \( k \neq -7 \), the only statement among the options that is correct if the system has a unique solution is \( k \neq 7 \).

Revision Table: System of Linear Equations Solutions

Additional Information: Consistency of Linear Systems

Systems of linear equations are broadly classified based on whether they have a solution or not. This property is called consistency.

• Consistent System: A system of equations is consistent if it has at least one solution. This includes systems with a unique solution and systems with infinitely many solutions.
• Inconsistent System: A system of equations is inconsistent if it has no solution. This happens when the equations represent lines that are parallel and distinct.

The concept of a unique solution falls under the category of consistent systems, specifically when the equations represent lines that intersect at a single point.