Problem 42 - Intermediate Algebra

Question

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} x=\frac{3}{2} y+5 \\ 2 x-3 y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent; no solution exists.

Step 1: Substitute for x

From the first equation, we have \( x = \frac{3}{2}y + 5 \). Substitute this expression for \( x \) into the second equation \( 2x - 3y = 8 \). So, replace \( x \) in the second equation to get: \( 2(\frac{3}{2}y + 5) - 3y = 8 \).

Step 2: Simplify the Equation

Distribute the 2 into the expression \((\frac{3}{2}y + 5)\). This gives \( 2 \times \frac{3}{2}y + 2 \times 5 = 3y + 10 \). So the equation becomes \( 3y + 10 - 3y = 8 \).

Step 3: Solve for y

Notice in the equation \( 3y + 10 - 3y = 8 \), the \(3y\) and \(-3y\) cancel each other out, leaving us with \( 10 = 8 \). This statement is false, meaning there is no solution for \( y \).

Step 4: Determine the Nature of the System

Since the final equation \( 10 = 8 \) is a contradiction, it indicates that the system of equations is inconsistent.

Key Concepts

Inconsistent Systems

An inconsistent system of equations is one in which no solutions exist. This happens when the lines represented by the equations are parallel. For example, if you visualize a system of two linear equations graphically, an inconsistent system results in two lines that never intersect.
Key characteristics of inconsistent systems include:

• Contradictory equations, such as getting a false statement like "10 = 8" when simplifying.
• Parallel lines with the same slope but different y-intercepts.

In our exercise, when attempting to solve the system, simplifying the equations leads to a contradiction (10 = 8), thereby, confirming the system is inconsistent. This indicates there is no possible set of values where both equations are true simultaneously.

Dependent Equations

Dependent equations are a different scenario where the equations in a system are essentially the same, meaning one is a multiple or transformation of the other.
In dependent systems, every solution of one equation is a solution of the other. This situation results in infinitely many solutions because the graphs of the equations coincide, showing the same line.
Signs of dependent equations include:

• The two equations simplify to the same line.
• One equation is a scalar multiple of the other.

In our example, the system was not dependent, as simplifying the two equations led to a contradiction rather than an identity. This confirms that they do not represent the same line.

Substitution Method

The substitution method is used for solving systems of equations by solving one of the equations for one variable and then substituting this expression into the other equation. This way, we effectively reduce the system into a single equation in one variable.
For this method:

• Solve one of the equations for one of the variables.
• Substitute the expression obtained into the other equation.
• Simplify and solve for the remaining variable.
• Back-substitute the solution into one of the original equations to find the other variable, if needed.

In our given problem, the first equation was solved for \( x \), and then \( x \) was substituted into the second equation. Simplifying led to a contradiction, revealing the system was inconsistent. This illustrates that even the substitution method sometimes reveals no solution due to the system's inherent inconsistency.