Problem 52 - A Graphical Approach to Precalculus with Limits

Question

For what value(s) of \(k\) will the following system of linear equations have no solution? infinitely many solutions? $$\begin{aligned}x-2 y &=3 \\\\-2 x+4 y &=k\end{aligned}$$

Step-by-Step Solution

Answer

The system has no solution when \( k = -6 \) and has infinitely many solutions when \( k = -6 \).

Step 1: Understand the concept of parallel lines

A system of linear equations will have no solution when the lines represented by the equations are parallel. This happens when the coefficients for the variables are proportional but the constant terms are not. Identify parallel line conditions for given equations.

Step 2: Express equations in standard form

The given equations are already in standard linear form: 1. \( x - 2y = 3 \) 2. \( -2x + 4y = k \). Notice the coefficients of terms in each equation.

Step 3: Equate coefficients for parallel condition

For the lines to be parallel, the coefficients and their ratios should be equal: \( \frac{1}{-2} = \frac{-2}{4} \). This simplifies to \( -\frac{1}{2} = -\frac{1}{2} \), which holds true.

Step 4: Identify the condition for no solution with different constants

If the lines are parallel, they must have different constant terms to ensure no solution. So, \( \frac{3}{k} eq \frac{1}{-2} \). Simplifying, we get \( k eq -6 \). Thus, the system has no solution if \( k = -6 \).

Step 5: Check for infinitely many solutions

For the lines to have infinitely many solutions, they must be the same line, requiring not just proportional coefficients but also identical constants. Thus, \( k = -6 \) in order for terms to be equal and identical.

Key Concepts

Parallel Lines and No Solution

A system of linear equations can have no solution if and only if the lines represented by the equations in the system are parallel. Parallel lines never meet; therefore, there can be no point that satisfies both equations simultaneously.
To determine if two lines are parallel, we look at the coefficients of the variables in each equation. The lines are parallel if:

• The coefficients of the variable terms are proportional, meaning the ratios of corresponding coefficients are equal, and
• The constant terms are not proportional. If they were, the lines would be the same, not parallel.

In the system given by the equations:\[ x - 2y = 3 \] \[ -2x + 4y = k \]We equate the ratios of the coefficients of the variables. For parallel lines:\[ \frac{1}{-2} = \frac{-2}{4} \]This simplifies to \(-\frac{1}{2} = -\frac{1}{2}\), which confirms the lines are parallel for any constant. To ensure a unique solution doesn’t exist, the constants must be inconsistent, giving us the no solution condition when \( k eq 6 \).

Infinitely Many Solutions

A system has infinitely many solutions when each equation represents the same line. This means that every point on one line is also a point on the other. For this condition to hold, the equations must be exactly equivalent.
The criteria for infinitely many solutions require:

• Proportional coefficients for the variables again, ensuring parallelism, and
• Identical constant terms, meaning the ratio between the constant terms must also match the ratio of the coefficients.

In our given system:\[ x - 2y = 3 \] \[ -2x + 4y = k \]We already have proportional coefficients:\[-\frac{1}{2} = -\frac{1}{2}\] Now, for the constant to be the same:\[-\frac{3}{k} = -\frac{1}{2} \]Solving this gives \( k = 6 \). Hence, when \( k = 6 \), both equations describe the same line, resulting in infinitely many solutions.

No Solution Condition

The no solution condition for a system of linear equations occurs when the two lines represented by the equations are parallel but not coincident. As mentioned, parallel lines are those with the same slope, but they will never meet unless they are exactly the same line.
To confirm a system has no solution:

• Ensure the ratios of the variable coefficients match, ensuring parallelism, and
• Verify that the constant terms are not proportional.

Returning to the example:\[ x - 2y = 3 \] \[ -2x + 4y = k \]The ratio of coefficients was \(-\frac{1}{2}\), proving the lines are parallel. When \( k eq 6 \), the constant term ratio differs, ensuring the lines are not the same line and never meet.
Thus, the system is assured of having no solution when \( k = 6 \), strongly guided by the different constant terms criterion.