Problem 27 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4. $$ \left\\{\begin{array}{l} x=3-2 y \\ 2 x+4 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent, with infinitely many solutions.

Step 1: Convert Equations to Slope-Intercept Form

First, let's convert each equation into the slope-intercept form, which is \( y = mx + b \). The first equation is already solved for \( x \) but needs to be rearranged to solve for \( y \).

Step 2: Sub-step for Equation 1

Start with \( x = 3 - 2y \). Rearrange to solve for \( y \) by adding \( 2y \) to both sides: \\[ 2y = 3 - x\] \ Divide each term by 2 to solve for \( y \): \\[ y = \frac{3-x}{2} \]

Step 3: Sub-step for Equation 2

For \( 2x + 4y = 6 \), solve for \( y \) by subtracting \( 2x \) from both sides: \\[ 4y = 6 - 2x \] \ Now, divide everything by 4: \\[ y = \frac{6 - 2x}{4} \] \ Simplify: \\[ y = \frac{3}{2} - \frac{x}{2} \]

Step 2: Graph Both Equations

Now that both equations are in slope-intercept form, plot them on a coordinate plane. The first equation \( y = \frac{3-x}{2} \) is a line with a slope of \(-\frac{1}{2}\) and a y-intercept at 1.5. The second equation \( y = \frac{3}{2} - \frac{x}{2} \) also has a slope of \(-\frac{1}{2}\) and a y-intercept at 1.5.

Step 3: Analyze the Graphs

Once the lines are plotted, observe that they are identical. This means they are dependent, sharing all points in common, because their slopes and y-intercepts are the same, indicating the same line.

Step 4: Check for Dependence or Inconsistency

Since the two lines overlap completely on the graph, the given system is dependent. This means the system has infinitely many solutions, as both equations represent the same line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental concept when dealing with linear equations in algebra. It is expressed as \( y = mx + b \), where:

• \(m\) represents the slope of the line, indicating its steepness and direction. A positive slope means the line ascends, while a negative slope means it descends.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

Using this form, it's much easier to visualize and graph equations because you directly see the slope and the starting point on the graph.

When converting equations to this form, as seen in the exercise, it lets you graph them quickly on a coordinate plane for further analysis. This conversion is vital for comparing and analyzing lines, especially in determining relationships between lines such as parallelism or overlap.

Dependent Systems

Dependent systems occur when two or more equations in a system essentially represent the same line. This means every solution of one equation is also a solution of the other. Such systems arise when the slopes and y-intercepts of the equations are identical.

In our exercise, both equations \( y = \frac{3-x}{2} \) and \( y = \frac{3}{2} - \frac{x}{2} \) were converted to slope-intercept form and found to have the same slope and y-intercept, which is symptomatic of a dependent system.

Identifying dependent systems graphically means observing that the lines overlap perfectly on the graph. As a graphing strategy, when you notice this overlap, it's a clear sign that one line can be used to predict outcomes for both equations, reinforcing that they are fully dependent.

Infinite Solutions

When a system of equations is dependent, as explained earlier, it has an infinite number of solutions. This is because each point on the overlapping lines serves as a valid solution for both equations.

In practical terms, what this means is:

• There is no single solution or intersection point, but rather a multitude of points where the two lines coincide.
• Any point \((x, y)\) that satisfies one equation will satisfy the other.

In our specific case, since the lines from both equations entirely overlap, each point on this line (and there are infinitely many) serves as a solution. Understanding infinite solutions helps in recognizing how settings with common variables can be modeled by the same scenario or rule, indicated by the line they share.