Problem 61 - Precalculus with Limits

Question

Finding the Value of a Constant In Exercises 61 and \(62,\) find the value of \(k\) such that the system of linear equations is inconsistent. $$ \left\\{\begin{aligned} 4 x-8 y &=-3 \\ 2 x+k y &=16 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The value of k that makes the system of equations become inconsistent is \( k = -4 \).

Step 1: Convert both equations into slope-intercept form \( y = mx + b \)

The first equation is: \( 4x - 8y = -3 \). Divide each term of the equation by -8 to isolate \( y \), and we obtain \( y = 0.5x + 0.375 \). \n\n The second equation is: \( 2x + ky = 16 \). To isolate \( y \), we get \( y = -2/kx + 16/k \).

Step 2: Compare the slopes of two lines

To effect inconsistency (parallel lines), the slopes of two lines must be equal. So we get: \n\n \( 0.5 = -2/k \)

Step 3: Solve the single variable algebraic equation

The equation would be in the form: \( k = -2 / 0.5 \). Solving this equation will give us the value of \( k \).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental when you're dealing with linear equations. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) signifies the y-intercept, which is the point where the line crosses the y-axis. To convert a linear equation into this form, you need to isolate \( y \) on one side of the equation.

Take for example the given system of equations. By manipulating the first equation \( 4x - 8y = -3 \), we can divide all terms by -8 to reconfigure it into slope-intercept form, yielding \( y = 0.5x + 0.375 \). This method is particularly useful as it immediately shows the rate at which \( y \) changes with \( x \), which is the slope, and where the line will intersect the y-axis. For the second equation \( 2x + ky = 16 \), dividing by \( k \) and then rearranging gives us a slope of \( -2/k \), which is important for identifying the type of system we're dealing with.

Parallel Lines

Lines are parallel when they have the same slope, meaning they will never intersect no matter how far they extend in both directions. This concept is crucial when solving for an inconsistent system, which is a set of equations representing parallel lines, thus having no solution.

In the case of our equations, after converting to slope-intercept form, you'll notice that both lines must have the same slope \( m \) to be parallel. From the given equations, the slope of the first line is \( 0.5 \), and we manipulated the second equation to express its slope as \( -2/k \). Setting these two slopes equal to each other because parallel lines have equal slopes allows us to solve for the value of \( k \) that would make the system inconsistent—that is, make the lines parallel.

Single Variable Algebraic Equation

The single variable algebraic equation is a foundational element of algebra that involves numbers and variables. It's often written in the form of \( ax + b = 0 \), where \( a \) and \( b \) are constants. To solve for the unknown variable \( x \), you would execute operations to isolate \( x \) on one side of the equation.

In the context of finding an inconsistent system, we already established that our slopes must match (from the parallel lines concept), leading us to the equation \( 0.5 = -2/k \). This is a simple single-variable equation where \( k \) is the variable. To isolate \( k \), we multiply both sides by \( k \) and then divide both sides by 0.5 to find that \( k \) must equal \( -4 \). Therefore, when \( k = -4 \), the two equations form an inconsistent system since they would represent parallel lines.