Problem 5 - Precalculus

Question

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{aligned} \frac{1}{2} x-\frac{1}{3} y &=-1 \\ 2 y-3 x &=6 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is consistent dependent with infinitely many solutions. Both equations represent the same line.

Step 1: Solve One Equation for One Variable

Let's begin by solving the first equation for the variable \( x \). The equation is \( \frac{1}{2}x - \frac{1}{3}y = -1 \). Multiply every term by 6 to eliminate fractions: \( 3x - 2y = -6 \). Next, solve for \( x \): \( 3x = 2y - 6 \), so \( x = \frac{2y-6}{3} \).

Step 2: Substitute into the Second Equation

Using the expression for \( x \) found in Step 1, substitute \( x = \frac{2y-6}{3} \) into the second equation \( 2y - 3x = 6 \). It becomes: \[ 2y - 3\left( \frac{2y-6}{3} \right) = 6 \]. Simplify by distributing and simplifying to get: \( 2y - 2y + 6 = 6 \), resulting in \( 0 = 0 \).

Step 3: Classify the System

Since substituting yielded the identity \( 0 = 0 \), this indicates that all values of \( x \) and \( y \) that satisfy one equation also satisfy the other equation. Therefore, the system is **consistent dependent**. Both equations represent the same line.

Step 4: Verify Algebraically

To verify, rearrange both equations to see if they represent the same line. First equation: \( 3x - 2y = -6 \) can be written as \( y = \frac{3}{2}x + 3 \). Second equation: \( 2y - 3x = 6 \) rearranges to \( y = \frac{3}{2}x + 3 \). Both equations represent the same line.

Step 5: Verify Graphically

Graphing the equations can also confirm this result. Plot \( y = \frac{3}{2}x + 3 \). Both equations will yield the same line on a graph, so there is no unique solution, confirming that the system is consistent dependent.

Key Concepts

Consistent Dependent Systems

In the world of solving linear systems, a consistent dependent system is one where all solutions of one equation make the other equation true as well. It's a special scenario where each equation in the system describes the same line when graphed. That's why such a system doesn't have a unique solution---instead, infinitely many solutions exist, and all these solutions lie along this line.
For the system of equations

• \(\frac{1}{2}x - \frac{1}{3}y = -1\)
• \(2y - 3x = 6\)

when solved using algebraic methods such as substitution or elimination, we find them to be consistent and dependent. This means rearranging both equations results in the same expression, confirming both describe the same line on a graph.
In such cases, there is no need to find separate solutions for each variable. Instead, any point on the line defined by one equation is a solution to the system.

Substitution Method

The substitution method is a popular technique to solve systems of equations. It involves expressing one variable in terms of another and using this expression to substitute in the other equation. Let's see how this method applies to our given system of equations:
First, take one of the equations and solve it for a variable. For instance, from \(\frac{1}{2}x - \frac{1}{3}y = -1\), we solve for \(x\), resulting in \(x = \frac{2y-6}{3}\).
Next, substitute this expression for \(x\) in the second equation \(2y - 3x = 6\). This will let you solve entirely in terms of one variable, typically simplifying down to an identity such as \(0 = 0\) in the case of a consistent dependent system.
This approach not only confirms the dependency of the equations but helps identify an infinite set of solutions since each solution of one variable automatically satisfies the other equation.

Elimination Method

The elimination method, another favorite among ways to solve linear systems, aims to eliminate one variable by adding or subtracting equations from each other. By making the coefficients of one of the variables the same or opposites, we can eliminate that variable from the system.
In our problem, we can multiply terms to avoid fractions in each equation:

• Multiply the first equation by 6 to get \(3x - 2y = -6\).
• The second equation is \(2y - 3x = 6\).

Now, add these equations together to eliminate both \(x\) and \(y\), resulting in the universal truth \(0 = 0\). This indicates all terms cancel out, confirming both lines are identical. Therefore, the elimination method reinforces that the system is consistent dependent, as both equations boil down to the same line.

Graphical Verification

Graphical verification is a highly visual and intuitive tool to confirm algebraic solutions of systems of equations. By graphing each equation, you can immediately see the relationship between them.
For our system, the equations

• \(y = \frac{3}{2}x + 3\)

show that both equations rearrange to give identical straight-line equations. When these graphs are plotted on the same axes, they do not form two intersecting lines but the same line. This graphically verifies the algebraic conclusion that the system is consistent dependent.
This method offers a clear picture: if the lines coincide, it's consistent dependent. If they intersect at a single point, it's consistent independent. If the lines are parallel and distinct, it's inconsistent.