Problem 7 - 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. $$ \left\\{\begin{aligned} 3 y-\frac{3}{2} x &=-\frac{15}{2} \\ \frac{1}{2} x-y &=\frac{3}{2} \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent, meaning it has no solution.

Step 1: Simplify the equations

First, simplify both equations to make calculations easier. The first equation is already arranged, so we'll rewrite it as it is: \[ 3y - \frac{3}{2}x = -\frac{15}{2} \]. For the second equation, we'll multiply through by 2 to eliminate the fractions: \[ x - 2y = 3 \]. Now, our system of equations is: \[ \begin{aligned} 3y - \frac{3}{2}x &= -\frac{15}{2} \ x - 2y &= 3 \end{aligned} \]

Step 2: Solve for x or y using substitution

We'll solve the second equation for \(x\). Rearrange \(x - 2y = 3\) to \(x = 2y + 3\). This expression for \(x\) will be substituted into the first equation.

Step 3: Substitute and solve for y

Substitute \(x = 2y + 3\) into the first equation: \[ 3y - \frac{3}{2}(2y + 3) = -\frac{15}{2} \]. Simplify: \[ 3y - 3y - \frac{9}{2} = -\frac{15}{2} \]. This results in \( -\frac{9}{2} = -\frac{15}{2}\), which simplifies to a false statement.

Step 4: Analyze the result

The false statement \( -\frac{9}{2} = -\frac{15}{2}\) indicates that there is no solution to the system. This suggests that the system is inconsistent.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another. Once the expression is found, it is substituted into another equation to find the value of one variable.
In this case, we have the system of equations: \[ \begin{aligned} 3y - \frac{3}{2}x &= -\frac{15}{2} \ x - 2y &= 3 \end{aligned} \]. To use the substitution method, we'll solve the second equation for \(x\). This equation becomes \(x = 2y + 3\).

This expression is then substituted into the first equation, replacing \(x\). This results in only one variable in this equation, allowing us to solve for \(y\). However, this approach ultimately led us to a false statement, indicating no solution. This shows that sometimes, substitution reveals the system's nature – inconsistent in this scenario.

Elimination Method

The elimination method involves manipulating equations in a system to eliminate one of the variables, making it easier to solve for the remaining variables.
This method usually involves multiplying equations so that one of the variables has the same coefficient (with opposite signs) in both equations, allowing for cancellation by addition or subtraction.
While the focus in the given problem was on substitution, the elimination method can be valuable for systems that lend themselves more easily to these types of adjustments. For instance, multiplying one or both equations by a constant could help align coefficients to directly eliminate a variable.

• It is important to note that this method helps highlight when there is no solution (inconsistent system) or infinitely many solutions (dependent system).

Consistent and Inconsistent Systems

When we talk about systems of equations, it's important to classify them as consistent or inconsistent. A consistent system has at least one solution, meaning the equations intersect at one point (independent solution) or are the same line (dependent solution).
An inconsistent system, like the one we encountered with our substitution method, has no solution. This usually happens when the lines represented by the equations are parallel and never intersect.
This classification is key in determining the best approach to solving the system and understanding the implications of algebraic manipulations that lead to clear contradictions, such as a false statement indicating inconsistency.

Graphical Method for Solving Equations

The graphical method involves plotting each equation in a system on the same coordinate plane and identifying points of intersection.
This visual representation provides an intuitive understanding of the solution or lack thereof. When equations intersect at a single point, the system is consistent and independent. If they coincide entirely, the system is dependent and consistent. If they are parallel and never meet, as found in our exercise, the system is inconsistent.

• Although not needed for algebraic proof, graphing supports the understanding of why a system might have no solutions.
• It offers a clear picture of geometric relationships between equations.

The graphical method thus complements algebraic techniques and confirms results visually.