Problem 24 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(4 x-2 y=6\) \(6 x-3 y=9\)

Step-by-Step Solution

Answer

The system is consistent and dependent; the equations represent the same line.

Step 1: Convert Equations to Slope-Intercept Form

First, express the equations in slope-intercept form (\(y = mx + b\)). Start with the first equation: \[4x - 2y = 6\] Solve for \(y\): \[-2y = -4x + 6\] Divide all terms by \(-2\): \[y = 2x - 3\] Now, convert the second equation: \[6x - 3y = 9\] Solve for \(y\): \[-3y = -6x + 9\] Divide all terms by \(-3\): \[y = 2x - 3\].

Step 2: Graph the Equations

Draw the graph for both lines. The first equation \(y = 2x - 3\) and the second equation \(y = 2x - 3\) are identical, so you plot just one line with a y-intercept of \(-3\) and a slope of \(2\). This means for every unit you move right on the x-axis, you move up 2 units on the y-axis.

Step 3: Analyze the Graph

After plotting the graphs, notice that the two lines are the same. Since they are identical, they overlap each other completely on the graph.

Step 4: Determine the Nature of the System

Since the two equations describe the same line on the graph, the system of equations has infinitely many solutions. Therefore, the system is consistent and dependent.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations, making it easy to identify the slope and the y-intercept of a line. It is represented as \( y = mx + b \), where:

• \( m \) indicates the slope of the line. This tells us how steep the line is—the rate of increase or decrease of \( y \) with respect to \( x \).
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert any linear equation to this form, solve for \( y \). For example, given \( 4x - 2y = 6 \), rearrange and simplify to isolate \( y \), resulting in \( y = 2x - 3 \). This tells us that the slope is 2, meaning for every move 1 unit to the right, the line goes up 2 units. The line crosses the y-axis at \( -3 \). This form makes graphing straightforward and analyzing relationships between variables intuitive.

Consistent and Dependent System

A consistent and dependent system of equations occurs when two or more equations represent the same line. This means they share all their solutions, resulting in infinitely many solutions.

• Consistent means that there exists at least one set of solutions that satisfies all equations.
• Dependent indicates that the equations are not independent. Essentially, one is a multiple or transformation of the other, leading to the same graph.

Using the equations from the original exercise, both expressions convert to the same slope-intercept form: \( y = 2x - 3 \). Therefore, they graph as one line—the identity of each equation reinforces this. Since the same line is plotted twice, consistency and dependency characterize this system. Such systems are perfect examples of equations that are interrelated in a way that they do not introduce new solutions but confirm existing ones.

Graphing Linear Equations

Graphing linear equations involves plotting lines on a coordinate plane based on their slopes and y-intercepts. This visual representation helps in understanding the nature of systems of equations

• Slope \((m)\): Indicates the tilt of the line. A positive slope rises, a negative slope falls, and a zero slope forms a horizontal line.
• Y-Intercept \((b)\): The starting point of the line on the y-axis. It is crucial for initially placing the line on a graph.

To graph an equation in slope-intercept form, begin at the y-intercept, \( b \), and use the slope, \( m \), to determine the next point. For the equation \( y = 2x - 3 \):

• Start at \( -3 \) on the y-axis.
• From this point, follow the slope: for every 1 unit you move to the right, move 2 units up.

Continue plotting points following this pattern. Once plotted, draw a line through these points to complete the graph. This process visually demonstrates the relationship between the variables described in each equation.