91Ó°ÊÓ

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.

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.

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.