Problem 38 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The system is consistent and independent.

Step 1: Rewrite the equations in slope-intercept form

The slope-intercept form of an equation is \( y = mx + b \). We first modify both equations to this form. Start with the first equation, \( 2y - 4x = 3 \). Solve for \( y \):\[ 2y = 4x + 3 \]Divide the entire equation by 2:\[ y = 2x + \frac{3}{2} \]. Now, the second equation: \( \frac{4}{3}x - y = -2 \). Solve for \( y \):\[ y = \frac{4}{3}x + 2 \].

Step 2: Identify the slope and y-intercept

The first equation \( y = 2x + \frac{3}{2} \) has a slope \( m = 2 \) and y-intercept \( b = \frac{3}{2} \). The second equation \( y = \frac{4}{3}x + 2 \) has a slope \( m = \frac{4}{3} \) and y-intercept \( b = 2 \).

Step 3: Graph the equations

Plot both lines on a coordinate plane. For the first equation, start at the y-intercept \( \left( 0, \frac{3}{2} \right) \) and use the slope to find another point: rise 2, run 1. For the second equation, start at the y-intercept \( \left( 0, 2 \right) \) and use the slope: rise 4, run 3.

Step 4: Analyze the graph

The lines from the equations \( y = 2x + \frac{3}{2} \) and \( y = \frac{4}{3}x + 2 \) intersect at one point. This indicates the system of equations has exactly one solution.

Step 5: Classify the System

Since the two lines intersect at one point, the system of equations is classified as consistent and independent.

Key Concepts

Graphing Equations

Graphing equations involves creating a visual representation of an algebraic expression on a coordinate plane. Each equation can be represented as a line, and plotting this line can help us understand the relationship between variables.
To graph an equation, first ensure it’s in a form that’s easy to interpret, such as the slope-intercept form. Plot the y-intercept, which is the starting point of the line. Then, use the slope to determine the direction and steepness of the line.

• Find the y-intercept on the y-axis.
• Use the slope to find additional points by "rising" and "running" according to the values.
• Draw the line through the points.

Graphing allows us to visually find intersections between lines, which signify solutions to a system of equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \). This format makes it easy to identify two crucial parts of a line: the slope and the y-intercept.
The slope \( m \) tells us how steep the line is and gives the rate of change between the variables. It’s calculated as the "rise over run" or the ratio of the vertical change to the horizontal change between two points on the line.
The y-intercept \( b \) is the point where the line crosses the y-axis. This is where the value of \( x \) is zero.

• Easy to read and plot.
• Quickly shows the growth rate and starting point.

Understanding this form simplifies graphing and analyzing linear equations.

Consistent Systems

A system of equations is consistent if at least one set of values exists that satisfies all equations simultaneously. In other words, their graphs will intersect with each other, and at that intersection point, the coordinates provide a solution to the system.
There are further classifications within consistent systems:

• Independent: These systems have exactly one solution, where the lines intersect at a single point.
• Dependent: Each equation represents the same line, hence infinite solutions.

Recognizing a system as consistent helps determine if solutions are possible and the intersection of the graphed lines provides these solutions.

Independent Systems

An independent system of equations is one where the equations represent lines that intersect at exactly one point. This means the system has a single, unique solution.
In the graph, this point of intersection represents the values of the variables that satisfy both equations. An independent system is always consistent as it results in a proper intersection.

• Unique solution: One intersection point.
• Forms distinct lines.

Identifying an independent system ensures that the problem-solving results in a clear and singular answer, as opposed to dependent or inconsistent systems where solutions are either infinite or nonexistent.