Problem 10 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(x-2 y=8\) \(\frac{1}{2} x-y=4\)

Step-by-Step Solution

Answer

The system is consistent and dependent.

Step 1: Rewrite first equation in slope-intercept form

The first equation is given as \(x - 2y = 8\). To rewrite it in slope-intercept form (\(y = mx + b\)), solve for \(y\):Subtract \(x\) from both sides:\[ -2y = -x + 8 \]Divide every term by \(-2\):\[ y = \frac{1}{2}x - 4 \].

Step 2: Rewrite second equation in slope-intercept form

The second equation is \(\frac{1}{2}x - y = 4\). Again, solve for \(y\):Subtract \(\frac{1}{2}x\) from both sides:\[ -y = -\frac{1}{2}x + 4 \]Multiply every term by \(-1\):\[ y = \frac{1}{2}x - 4 \], which is the same equation as in Step 1.

Step 3: Compare the equations

Both equations were simplified to \(y = \frac{1}{2}x - 4\). Since they are identical, the system of equations represents the same line.

Step 4: Classify the system of equations

Since both equations represent the same line, the system of equations is consistent and dependent.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to easily understand the line in terms of its slope and y-intercept. It's written as:\[y = mx + b\]Where:

• \( m \) is the slope of the line, indicating its steepness and direction.
• \( b \) is the y-intercept, which tells you where the line crosses the y-axis.

This form is especially useful for graphing because you can quickly plot the y-intercept on the graph and use the slope to find other points on the line. The slope describes how much \( y \) changes for a one-unit change in \( x \). For instance, in the exercise, transforming both equations into slope-intercept form helps immediately see they are actually the same line: \( y = \frac{1}{2}x - 4 \). This indicates that they have identical slopes and y-intercepts.

Consistent Systems

A system of equations is termed as consistent if it has at least one solution. These systems can be either independent or dependent.

• A consistent and independent system has exactly one solution, meaning the lines intersect at a single point.
• On the other hand, a consistent and dependent system means the equations indeed describe the same line, leading to infinitely many solutions.

In the exercise provided, after transforming both equations into slope-intercept form, they become identical, pointing to a consistent and dependent system because every point on the line is a solution.

Dependent Systems

Dependent systems of equations occur when multiple equations describe the same line. In other words, every solution of one equation is also a solution of another. This normally leads to infinitely many solutions, unlike independent systems with unique intersections.

• For example, if two lines have the same slope and y-intercept, they are dependent, as they overlap completely.
• In the provided problem, both equations, once simplified, resulted in \( y = \frac{1}{2}x - 4 \).

This is a classic example of a dependent system, where both routes lie perfectly over each other on a graph.

Solving Equations

Solving a system of equations can be done through various methods, including graphing, substitution, or elimination. However, graphing offers a visual solution, showing where lines intersect.

• Start by converting equations to slope-intercept form, like in the exercise, to easily plot them.
• If two lines appear superimposed, as seen here, they represent a dependent system with infinite solutions.
• In contrast, if you find a clear intersection point, you've identified an independent, consistent system with a unique solution.

The process of solving these equations reveals critical relationships between the lines, ultimately helping to classify the system correctly.