Problem 39 - 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. $$ \begin{array}{rr} x-2 y= & -6 \\ -2 x+y= & 6 \end{array} $$

Step-by-Step Solution

Answer

The system is consistent and independent with a solution at (-2, 2).

Step 1: Rewrite the Equations

First, rewrite each of the given equations in the slope-intercept form, which is \(y = mx + b\). For the first equation, \(x - 2y = -6\), solve for \(y\):\[\begin{align*} x - 2y &= -6 \ -2y &= -x - 6 \ y &= \frac{1}{2}x + 3 \end{align*}\]For the second equation, \(-2x + y = 6\), solve for \(y\):\[\begin{align*} -2x + y &= 6 \ y &= 2x + 6 \end{align*}\]. Now, the equations are: \(y = \frac{1}{2}x + 3\) and \(y = 2x + 6\).

Step 2: Graph the Equations

Plot the lines represented by each equation on a coordinate plane: - Line 1 (\(y = \frac{1}{2}x + 3\)) has a slope of \(\frac{1}{2}\) and a y-intercept at (0, 3).- Line 2 (\(y = 2x + 6\)) has a slope of 2 and a y-intercept at (0, 6).Draw these lines to see where they intersect. The intersection point is the solution to the system of equations.

Step 3: Find the Intersection Point

To find the intersection point, set the two equations equal to each other since each equation represents the \(y\) value:\[\begin{align*}\frac{1}{2}x + 3 &= 2x + 6 \3 &= \frac{3}{2}x + 6 \\frac{3}{2}x &= -3 \x &= -2 \y &= 2(-2) + 6 = 2\end{align*}\]Thus, the intersection point is \((-2, 2)\).

Step 4: Verify the Solution

Substitute \(x = -2\) and \(y = 2\) back into the original equations to verify: - For \(x - 2y = -6\): \[-2 - 2(2) = -2 - 4 = -6\], which is correct.- For \(-2x + y = 6\): \[ -2(-2) + 2 = 4 + 2 = 6\], which is also correct.The point satisfies both equations.

Step 5: Classify the System

Since the graph of the equations meets at exactly one point, the system is consistent. Each equation represents a different line, so the equations are independent.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting the lines represented by the equations on a coordinate plane. Each linear equation can be expressed in the slope-intercept form, which is:

• \(y = mx + b\)

where \(m\) represents the slope, and \(b\) is the y-intercept. The slope tells you how steep the line is and the direction it goes. The y-intercept is where the line crosses the y-axis.

In our exercise, we rewrote the equations into the slope-intercept form:

• \(y = \frac{1}{2}x + 3\) with slope \(\frac{1}{2}\) and y-intercept \(3\).


• \(y = 2x + 6\) with slope \(2\) and y-intercept \(6\).

Start by plotting the y-intercept on the graph. Then, use the slope to determine another point and draw the line. For example, with a slope of \(\frac{1}{2}\), from the y-intercept (0, 3), move up 1 unit and right 2 units to find another point. The intersection of these two lines represents the solution to the system of equations.

Consistent Systems

A consistent system of equations is one where there is at least one set of solutions that satisfy all the equations in the system. In terms of graphing, consistent systems are represented by lines that intersect at least once on a coordinate plane.

There are two types of consistent systems:

• Consistent and independent, where the lines intersect at a single point, indicating exactly one solution.


• Consistent and dependent, where the lines coincide entirely, indicating infinitely many solutions because they are essentially the same line.

In our example, the two lines intersect at the point \((-2, 2)\), indicating one solution. Thus, the system is consistent and independent.

Dependent and Independent Equations

Equations in a system can be categorized as dependent or independent based on their relationship with each other.

• Independent equations: These are equations that represent different lines on the graph. Each equation contributes uniquely to the system, leading to one single intersection point when graphed, which represents the unique solution of the system. In such cases, the set of equations is both consistent and independent. Our system is an example of independent equations as they intersect at \((-2, 2)\).


• Dependent equations: These involve cases where one equation can be derived from another, indicating they represent the same line on a graph. They lead to infinitely many intersection points because the graphs overlap completely. Such a system is consistent but dependent, as there is not a unique solution.

Understanding these concepts helps in classifying systems of equations accurately, aiding in predicting the number and nature of possible solutions.