Problem 35 - College Algebra with Modeling and Visualization

Question

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{array}{r} 2 x+y=3 \\ -2 x-y=4 \end{array} $$

Step-by-Step Solution

Answer

The system is inconsistent and independent as the lines are parallel and do not intersect.

Step 1: Convert Equations to Slope-Intercept Form

First, convert each of the given equations into slope-intercept form \(y = mx + b\).For the first equation, \(2x + y = 3\), solve for \(y\):\[ y = -2x + 3 \]For the second equation, \(-2x - y = 4\), solve for \(y\):\[ y = -2x - 4 \]

Step 2: Analyze Slopes and Y-Intercepts

Compare the slope and y-intercept of the two equations.Both equations have a slope of \(-2\), which indicates they are parallel.The y-intercepts \(b_1 = 3\) and \(b_2 = -4\) are different, confirming that the lines are distinct and parallel.

Step 3: Graph the Equations

Plot both equations on the same coordinate grid.The first line, \(y = -2x + 3\), crosses the y-axis at 3 and has a slope of \(-2\).The second line, \(y = -2x - 4\), crosses the y-axis at -4 and also has a slope of \(-2\).The lines are parallel and do not intersect.

Step 4: Determine the Consistency and Dependence of the System

Since the lines are parallel and do not intersect, the system does not have any solutions. This makes the system inconsistent. Additionally, because they are distinct lines (different y-intercepts), the equations are independent.

Key Concepts

Graphing Equations

Graphing equations is a fundamental method in solving systems of equations. By plotting lines on a coordinate plane, we can visually determine the relationships between them. Each equation represents a line, and the solution to a system of equations corresponds to the points where these lines intersect.

To graph a linear equation, it is usually in the form of the slope-intercept equation:

• The equation is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• The slope \(m\) indicates the steepness and direction of the line; a positive slope ascends from left to right, while a negative slope descends.
• The y-intercept \(b\) is the point where the line crosses the y-axis.

By graphing the equation \(y = -2x + 3\) you place a point at 3 on the y-axis, then follow the slope \(-2\) to mark another point. For each unit you move right, you descend 2 units. Similarly, graph \(y = -2x - 4\) by starting at -4 on the y-axis, and again using the slope \(-2\). Once both lines are drawn, you can visually inspect to find where (and if) they intersect. If they are parallel, as they are in this case, they will not meet.

Consistency of Equations

A system of equations can be classified as either consistent or inconsistent, which tells us about the number of solutions it has.

• A consistent system has at least one solution. The solution could be a single point where the lines intersect, or it could be infinitely many solutions if the lines overlap completely.
• An inconsistent system has no solutions. This happens when the lines are parallel and distinct, meaning they never intersect on the graph.

In the context of our given system, after converting and graphing both equations, we found that the lines are parallel and do not intersect. Each slope is equal (\(-2\)), while the y-intercepts (3 and -4) differ, meaning these lines will never meet on the plane. As a result, this system is inconsistent.

Dependent and Independent Equations

Understanding the dependency of equations in a system helps determine the nature of the solutions. A set of dependent equations means one equation can be derived from another, indicating they are essentially the same when graphed.

• Dependent equations have infinitely many solutions because they describe the same line, so they overlap completely on a graph.
• Independent equations are distinct and describe different lines. They can either intersect at a single point (consistent and independent) or never intersect at all if they are parallel (inconsistent and independent).

In the solved system, the lines were independent because they had different y-intercepts, despite having the same slope. Therefore, these were independent equations. Since the lines did not meet, they were inconsistent, confirming independence as they have no solutions compared to dependent equations.