Problem 63 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} x=-\frac{3}{2} y \\ 2 x=3 y-4 \end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Understand the System of Equations

We are given the system of equations: \(x = -\frac{3}{2} y\) and \(2x = 3y - 4\). We need to plot these equations on a graph to find their point of intersection, which will provide a solution to the system.

Step 2: Rewrite the Equations for Graphing

Rewrite each equation in terms of \(y\) to better facilitate graphing:1. From \(x = -\frac{3}{2} y\), solve for \(y\): \[y = -\frac{2}{3}x\] 2. From \(2x = 3y - 4\), solve for \(y\): \[3y = 2x + 4 \Rightarrow y = \frac{2}{3}x + \frac{4}{3}\]

Step 3: Graph the Lines

Using the equations from Step 2, graph the lines:1. For \(y=-\frac{2}{3}x\): This line passes through the origin (0,0) as the y-intercept is 0. It has a negative slope of -\(\frac{2}{3}\).2. For \(y = \frac{2}{3}x + \frac{4}{3}\): This line has a y-intercept at \(\frac{4}{3}\) and a positive slope of \(\frac{2}{3}\).

Step 4: Find the Intersection Point

Examine where the two lines intersect on the graph. It is the point where the \(y\)-values and \(x\)-values are the same for both equations.

Step 5: Analyze Solution

Since the graph shows that the lines have the same slope but different y-intercepts, they are parallel and will never intersect. This means the system has no solution.

Key Concepts

Graphing Systems

When faced with solving a system of equations graphically, it involves plotting each equation as a line on a coordinate plane. Each line represents all the solutions to that particular equation.
To start, equations are often rewritten in slope-intercept form \(y = mx + b\) for easier graphing, where \(m\) is the slope and \(b\) is the y-intercept. This format is intuitive for sketching lines.

The process of graphing involves:

• Identifying the y-intercept, which tells you where the line crosses the y-axis.
• Using the slope to determine the direction and steepness of the line, often moving up/down and left/right from the y-intercept.

By graphing both equations, the intersection point, if it exists, gives the \(x\) and \(y\) values that satisfy both equations. This is the solution to the system of equations.

Systems with No Solution

A system of equations can sometimes have no solution. Graphically, this occurs when the lines are parallel.
Parallel lines have the same slope but different y-intercepts, meaning they never meet at any point on the graph.
The system we're working with in this exercise exhibits this exact situation:

• Both equations, once simplified, have slopes of \(-\frac{2}{3}\) and \(\frac{2}{3}\), but different y-intercepts.
• This difference in y-intercepts means the lines run alongside each other without ever intersecting.

Such a system is labeled "inconsistent", indicating that there is no set of values for \(x\) and \(y\) that will satisfy both equations simultaneously.

Dependent and Independent Equations

Understanding whether equations are dependent or independent is crucial in solving them correctly.

**Independent Equations**
These are distinct equations that intersect at one unique point. Their graphs will cross at exactly one point unless they are parallel.

**Dependent Equations**
These equations represent the same line, even if they appear different at first glance. This means all their solutions overlap, and their graphs lie atop one another entirely.
However, our current system does not involve dependent equations since the lines are parallel (indicating no overlap).

Knowing whether equations are dependent or independent helps you anticipate the nature of the solution. It sets the stage for understanding if a system will have a single solution, no solution, or infinitely many solutions.