Problem 52 - Intermediate Algebra

Question

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} \frac{5}{3} x+2=2(y+6) \\ 3 y+5=\frac{5}{2}(x-4) \end{array}\right. $$

Step-by-Step Solution

Answer

The system is consistent and the equations are dependent, representing the same line.

Step 1: Simplify First Equation

Start by simplifying the first equation \( \frac{5}{3} x + 2 = 2(y + 6) \) by distributing the 2 on the right side. This gives us: \( \frac{5}{3} x + 2 = 2y + 12 \). Now, rearrange to express \( y \) in terms of \( x \): \( 2y = \frac{5}{3} x + 2 - 12 \) which simplifies to \( 2y = \frac{5}{3} x - 10 \). Divide both sides by 2: \( y = \frac{5}{6}x - 5 \).

Step 2: Simplify Second Equation

Now simplify the second equation \( 3y + 5 = \frac{5}{2}(x - 4) \). First, distribute the \( \frac{5}{2} \): \( 3y + 5 = \frac{5}{2}x - 10 \). Rearrange to express \( y \) in terms of \( x \): \( 3y = \frac{5}{2}x - 10 - 5 \) or \( 3y = \frac{5}{2}x - 15 \). Divide both sides by 3: \( y = \frac{5}{6}x - 5 \).

Step 3: Determine Consistency and Dependency

Both equations simplify to \( y = \frac{5}{6}x - 5 \). This indicates that the two equations are dependent, meaning they represent the same line. Since there are infinitely many solutions that satisfy both equations, the system is consistent but the equations are dependent.

Key Concepts

Dependent Equations

When we talk about **dependent equations**, we are referring to two or more equations that essentially describe the same line in a graph. In simpler terms, one equation can be rewritten as a multiple or a combination of the other.
This type of situation arises when trying to solve a system of linear equations, and it implies that the equations do not provide new information regarding the relationships between variables. In the example given, both equations simplify to the equation:

• \( y = \frac{5}{6}x - 5 \)

Because both equations come down to the same line, they are dependent. Dependent equations are interesting because they don't have a unique solution—instead, every point on the line \( y = \frac{5}{6}x - 5 \) is a solution to the system. This results in an **infinite number of solutions**.

Consistent Systems

A **consistent system** of equations is one that has at least one solution. This is an important aspect in solving linear systems, as it indicates the system is solvable to some extent.
There are two main types of consistent systems:

• Systems with exactly one solution, where the lines intersect at a single point.
• Systems with infinitely many solutions, where the lines are overlapping, as seen in the dependent equations like our discussed example.

In our case, since both equations simplify to the same line, the system is consistent with infinitely many solutions. This implies that each solution point on the line represents a valid solution for both original equations.

Solving Systems of Equations

**Solving systems of equations** involves finding all possible solutions that satisfy all equations simultaneously. There are several methods you might evaluate or employ, including:

• Substitution
• Elimination
• Graphing

For our given system:

• The substitution method worked efficiently, simplifying both equations to express \( y \) in terms of \( x \).
• We discovered that they both collapse to the same equation: \( y = \frac{5}{6}x - 5 \).

This revelation makes clear that solving these leads to acknowledging the dependence between them. As a result, by using substitution, we could effectively determine the consistency and dependency without the need to solve directly for a coordinate solution. Solving systems like these often requires patience and careful analysis but ultimately reveals valuable insights about their nature.