Problem 53 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Equations in Slope-Intercept Form

To graph the equations easily, rewrite them in slope-intercept form, which is \(y = mx + b\). For the first equation, \(4y - 2x = 4\), solve for \(y\):First, add \(2x\) to both sides to get \(4y = 2x + 4\). Then, divide every term by \(4\) to isolate \(y\):\[y = \frac{1}{2}x + 1\].For the second equation, \(y - \frac{1}{2}x = 1\), it is already in the form \(y = mx + b\):\[y = \frac{1}{2}x + 1\].

Step 2: Graph the Equations

Graph both equations on the same set of axes.- The first equation, \(y = \frac{1}{2}x + 1\), has a y-intercept at (0, 1) and a slope of \(\frac{1}{2}\), which means rise 1 unit for every 2 units to the right.- The second equation is exactly the same: \(y = \frac{1}{2}x + 1\), so it will have the same graph.Because both lines have the same slope and y-intercept, they are identical and coincide with each other.

Step 3: Determine the Nature of the System

Since both equations produce the same line, the system of equations is consistent and dependent. Consistent means that there is at least one solution, and dependent means the equations represent the same line, therefore, there are infinitely many solutions.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting lines on a coordinate plane. Linear equations can be expressed in various forms, but the most convenient form for graphing is the slope-intercept form. To graph an equation like this, you need two main components:

• The y-intercept: This is where the line crosses the y-axis. It is represented by the 'b' in the equation.
• The slope: This indicates the steepness of the line and the direction it tilts. It's represented by 'm' in the equation, where a positive slope rises and a negative slope falls as it moves across the coordinate plane.

To graph the equation, start by plotting the y-intercept on the graph. Then use the slope to find another point on the line. For example, with a slope of \(\frac{1}{2}\), starting at the y-intercept, go up 1 unit and right 2 units to find a new point. Connect these points to form a line. Graphing multiple equations helps in visualizing solutions where their graphs intersect or overlap.

Consistent and Dependent Systems

A system of equations can be described based on how their graphs relate to each other. In the case of consistent and dependent systems, the concept is straightforward. A consistent system is one where at least one solution exists for the equations involved. On the other hand, dependence refers to the equations having the same graph. This means:

• Consistent: The graphs of the equations share at least one common point, indicating at least one solution exists.
• Dependent: If the equations produce identical lines, they coincide and also have infinitely many solutions. Essentially, each solution of one equation is a solution of the other.

In the given exercise, both equations simplify to the same line form. Hence, all points on this line are solutions, demonstrating that the system is both consistent and dependent. These conditions showcase that the equations don't just overlap at specific points but align perfectly along the entire line.

Slope-Intercept Form

The slope-intercept form of a linear equation is a simple and informative way to express lines. It's written as \(y = mx + b\), where:

• 'm' denotes the slope: It describes how steep the line is. A larger value makes the line steeper, and the sign indicates direction.
• 'b' is the y-intercept: It indicates where the line crosses the y-axis. This is the point \((0, b)\) on the graph.

To convert any linear equation to slope-intercept form, solve for \(y\) to isolate it on one side of the equation. This often makes graphing easier. For instance, in the given exercise, both original equations were rearranged into this form:

• The first equation, initially written as \(4y - 2x = 4\), was simplified to \(y = \frac{1}{2}x + 1\).
• The second one already matched \(y = \frac{1}{2}x + 1\).

By using the slope-intercept form, you can quickly identify both the slope and y-intercept, allowing for straightforward graphing and comparison of equations.