Problem 61 - 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.) $$ \left\\{\begin{array}{l} x+3 y=6 \\ y=-\frac{1}{3} x+2 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent and has infinitely many solutions.

Step 1: Rewrite the Equations

The system of equations to solve is given as: \( x + 3y = 6 \) and \( y = -\frac{1}{3}x + 2 \). The second equation is already in slope-intercept form (\(y = mx + b\)). Let's rewrite the first equation in slope-intercept form as well: \( y = -\frac{1}{3}x + 2 \). As we can see, upon solving \(x + 3y = 6\) for \(y\), we get \( 3y = -x + 6 \) which simplifies to \( y = -\frac{1}{3}x + 2 \).

Step 2: Identify Slope and Intercept

In slope-intercept form \( y = mx + b \), \(m\) is the slope and \(b\) is the y-intercept. Both equations are \( y = -\frac{1}{3}x + 2 \), hence they have the same slope \( m = -\frac{1}{3} \) and the same y-intercept \( b = 2 \).

Step 3: Graph the Equations

To graph the equations, plot the y-intercept (0, 2) on a coordinate plane for both lines. Since the slope is \(-\frac{1}{3}\), move down 1 unit and right 3 units. Repeat this to plot the second point and draw the lines for both equations.

Step 4: Analyze the Graph

Since the equations represent the same line, when graphed, they overlap completely. Both equations represent the same linear equation on the graph.

Step 5: State the Nature of the System

The system is dependent because both equations describe the same line. This means there are infinitely many solutions, as every point on the line satisfies both equations.

Key Concepts

Dependent Systems

A dependent system of equations occurs when two or more equations describe the same line. This happens when the equations are essentially multiples of one another. In the exercise, both equations are ultimately equivalent, with identical slopes and y-intercepts, meaning they overlap completely when graphed.

• If the lines described by the equations are perfectly overlapping, then each point on the line is a solution to both equations.
• To identify a dependent system, check if the equations can be transformed into each other with simple mathematical alterations, like multiplying by a constant.

In a dependent system, because the equations represent the same line, the graph of one is indistinguishable from the graph of the other, thus depicting an endless number of solutions. This is quite different from consistent systems, which intersect at a single point, and inconsistent systems, where the lines will never meet.

Slope-Intercept Form

The slope-intercept form of a linear equation is particularly useful in graphing because it directly communicates the slope and the y-intercept of the line. The general form is given by:
\[y = mx + b\]
Here, \(m\) represents the slope, and \(b\) represents the y-intercept—the point where the line crosses the y-axis.

• The slope \(m\) describes how steep the line is and in which direction it moves. A positive slope means the line moves upwards as it goes from left to right, while a negative slope moves downwards.
• The y-intercept \(b\) is easy to identify on a graph, marking exactly where the line meets the y-axis.

In the given solution, the lines had a slope of \(-\frac{1}{3}\) and a y-intercept at 2. This made graphing straightforward because knowing the starting point (the y-intercept) and the direction and steepness of the line (the slope) allows students to accurately draw the line on a coordinate plane.

Infinitely Many Solutions

A system of equations has infinitely many solutions when the equations are dependent. This means that instead of merely having a single intersection point, every point on the line is a solution simultaneously.

• The occurrence of infinitely many solutions is typical for dependent systems, where both equations reduce to the same equation and signify the same geometrical line.
• Given infinite solutions, the system doesn't just apply to one single point but rather encompasses a continuous line of solutions that can be graphed and visualized as one single line where all points are solutions.

For students, recognizing this key feature is essential because it entirely changes how a system is understood and interpreted. Rather than looking for one particular set of \((x, y)\) values, every possible pair along the line is valid, effectively creating an endless list of solutions.