Problem 16 - Intermediate Algebra

Question

For Problems \(1-16\), use the graphing approach to determine whether the system is consistent, the system is inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. (Objective 1) $$ \left(\begin{array}{l} y=-2 x \\ 3 x-y=0 \end{array}\right) $$

Step-by-Step Solution

Answer

The system is consistent with the solution (0, 0).

Step 1: Understanding the Equations

We have two equations: \( y = -2x \) and \( 3x - y = 0 \). Our task is to analyze these equations graphically to determine if the system is consistent (has a solution), inconsistent (has no solutions), or dependent (equations represent the same line).

Step 2: Rewrite the Second Equation in Slope-Intercept Form

The second equation is given as \( 3x - y = 0 \). Solve for \( y \) to express it in the slope-intercept form \( y = mx + b \). Thus, add \( y \) to both sides to get \( y = 3x \). Now, we have two equations in slope-intercept form: \( y = -2x \) and \( y = 3x \).

Step 3: Graph the Equations

Plot the equations \( y = -2x \) and \( y = 3x \) on the same coordinate plane. The line \( y = -2x \) has a negative slope of \(-2\), passing through the origin (0,0), while \( y = 3x \) has a positive slope of \(3\), also passing through the origin.

Step 4: Analyze the Graph

Observe the points of intersection of the graphs of the equations. Since both lines intersect at exactly one point, (0, 0), this indicates the lines are not parallel and meet at only one point. This implies the system is consistent.

Step 5: Verify the Solution

The point of intersection found is \((0, 0)\). Substitute \(x = 0\) and \(y = 0\) back into both equations to verify: \(y = -2(0) = 0\) for the first equation, and \(3(0) - 0 = 0\) for the second equation. Both are satisfied, confirming the solution.

Key Concepts

Consistent Systems

When discussing systems of equations, a consistent system refers to a set of equations having at least one solution. Consistent systems can be further divided into those that have exactly one solution, called independent systems, and those with infinitely many solutions, known as dependent systems.

• Independent systems: These systems have distinct equations that intersect at exactly one point. This means the slopes of the lines are different, and they cross each other on the graph.
• Dependent systems: Though still consistent, these involve equations where one is a multiple of the other, causing the lines to overlap entirely on the graph, leading to infinitely many solutions.

In the given exercise, the equations formed a consistent system because both lines intersected at exactly one point, (0, 0). This confirms that a solution exists for the system.

Inconsistent Systems

Inconsistent systems of equations do not have any solutions. This occurs when the equations form lines that are parallel, as they never intersect.

• Parallel lines: When lines have the same slope but different y-intercepts, they run parallel on the graph and never meet.
• No solutions: Because the lines don't intersect, there is no common point that satisfies both equations.

Understanding inconsistent systems is important because it helps you recognize when equations are set up such that they cannot be resolved with a common solution. In our exercise, this was not the case as the system was consistent, with the lines crossing at (0, 0).

Dependent Systems

Dependent systems are a type of consistent system where the equations result in lines that lie exactly on top of each other, resulting in infinitely many solutions. In essence, each solution to one equation is also a solution to the other, making the system dependent.

• Same line representation: In dependent systems, each equation is a scalar multiple of the other, which means they are essentially the same.
• Infinite solutions: Because the lines overlap completely, there are countless solutions along the line.

For the equations in the exercise, a dependent system would mean rewriting one equation as a multiple of the other. However, this was not observed since these lines intersected only at one distinct point.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is particularly helpful as it allows for easy graphing and comparison of line slopes to analyze relationships between lines.

• Slope \(m\): Indicates the steepness of the line. A positive slope suggests an upward angle, whereas a negative slope suggests a downward one.
• Intercept \(b\): Where the line crosses the y-axis. Changing this value will shift the line up or down without altering its slope.

For our exercise, converting the second equation into slope-intercept form simplified the graphing process as it allowed us to compare the slopes directly, confirming the two lines' intersection at one point only.