Problem 38 - 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. $$ \begin{array}{l} -x+2 y=3 \\ 3 x-y=1 \end{array} $$

Step-by-Step Solution

Answer

The solution is \((1, 2)\). The system is consistent and independent.

Step 1: Rewrite the equations in slope-intercept form

Start by rewriting each equation in the form \( y = mx + b \).For the first equation, \(-x + 2y = 3\):First, add \(x\) to both sides:\[2y = x + 3\]Next, divide everything by 2:\[y = \frac{x}{2} + \frac{3}{2}\]For the second equation, \(3x - y = 1\):Add \(y\) to both sides, then subtract 1:\[3x - 1 = y\]Thus, \(y = 3x - 1\).

Step 2: Graph the equations

Use the slope-intercept form to plot each equation on a graph.For the first equation, \(y = \frac{x}{2} + \frac{3}{2}\), the y-intercept is \(\frac{3}{2}\), and the slope is \(\frac{1}{2}\).For the second equation, \(y = 3x - 1\), the y-intercept is \(-1\) and the slope is \(3\).Plot the y-intercepts and use the slopes to plot more points. Draw the two lines on the same coordinate plane.

Step 3: Identify the point of intersection

Observe the graph to see where the lines intersect.The lines intersect at a point that solves both equations. By examining the drawn graph, the intersection point should be at \((1, 2)\).

Step 4: Verify the solution

Substitute \((1, 2)\) into both equations to ensure it satisfies them.For \(-x + 2y = 3\):\[-1 + 2(2) = 3\]\[-1 + 4 = 3\]which is true.For \(3x - y = 1\):\[3(1) - 2 = 1\]\[3 - 2 = 1\]which is also true.Both equations are satisfied, confirming \((1, 2)\) is the correct solution.

Step 5: Classify the system as consistent or inconsistent

Since the lines intersect at exactly one point, the system of equations is consistent. Because the lines are not parallel and intersect at exactly one point, the equations are independent.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting straight lines on a coordinate plane. For each equation, we want to rewrite it in the slope-intercept form, which is given by \( y = mx + b \). This format clearly shows the slope \( m \) and the y-intercept \( b \), which are essential for graphing.

• Slope \( m \): Indicates the steepness of the line, calculated as the change in \( y \) (rise) over the change in \( x \) (run).
• Y-intercept \( b \): The point where the line crosses the y-axis.

For example, take the equation \( y = \frac{x}{2} + \frac{3}{2} \). Here, the slope is \( \frac{1}{2} \) meaning for every 2 units we move horizontally, the line moves 1 unit vertically. The y-intercept is \( \frac{3}{2} \), the starting point on the y-axis.Plotting involves starting at the y-intercept and following the slope to find another point on the line. Repeat this for the second equation \( y = 3x - 1 \), where the slope is 3, and the y-intercept is -1. After plotting both lines, look for where they intersect. This point is the solution of the system if it exists.

Consistent and Inconsistent Systems

Systems of equations are either consistent or inconsistent. This classification depends on whether the system of equations has at least one solution.

• Consistent Systems: These have at least one solution. This means the lines representing the equations on a graph will intersect at some point.
• Inconsistent Systems: These do not have any solutions. On a graph, the lines will be parallel, never meeting each other.

In our example, the lines representing the system intersect at the point (1, 2), confirming that the system is consistent. By definition, any system where the equations' graphical representations meet at any single or multiple points is consistent. In such systems, solutions exist that satisfy both equations.

Dependent and Independent Equations

When dealing with consistent systems, equations can either be dependent or independent. This distinction explains how equations relate to each other graphically.

• Dependent Equations: In these, one equation can be derived from the other. Graphically, it means the equations represent the same line, resulting in infinitely many solutions – every point on the line is a solution.
• Independent Equations: These are distinct and not derived from each other. They intersect at only one point, providing exactly one solution to the system.

In our context, since the system intersects at exactly one point, the equations are independent. Each equation represents a different line, and as these lines cross at a single intersection, they share only that one solution set: (1, 2). Understanding this distinction helps determine how equations relate and the nature of the solutions they provide.