Problem 16 - Precalculus: Functions and Graphs

Question

Solve the system. $$\left\\{\begin{array}{r} x-5 y=2 \\ 3 x-15 y=6 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \((x, y) = (5y + 2, y)\).

Step 1: Write the System of Equations

The given system of equations is:1. \( x - 5y = 2 \)2. \( 3x - 15y = 6 \)

Step 2: Notice the Relationship Between Equations

Observe that the second equation \( 3x - 15y = 6 \) is actually a multiple of the first equation \( x - 5y = 2 \). If you multiply the first equation by 3, you get:\( 3(x - 5y) = 3 \times 2 \)This simplifies to:\( 3x - 15y = 6 \), which is identical to the second equation.

Step 3: Conclusion - Dependent System

Since the second equation is simply a multiple of the first, the system of equations is dependent, meaning there are infinitely many solutions. The two equations are effectively the same line.

Step 4: Express General Solution

We can express the general solution in terms of one variable. If we solve the first equation for \( x \), we have:\( x = 5y + 2 \)Thus, the solution set is all pairs \( (x, y) \) such that \( x = 5y + 2 \).

Key Concepts

Dependent Systems

A system of linear equations is called "dependent" if the equations are not truly independent of each other. This means one equation can be derived from another by multiplication or any simple algebraic manipulation. In our given system, \( x - 5y = 2 \) and \( 3x - 15y = 6 \), the second equation is merely the first equation multiplied by 3, indicating that these two lines are actually the same geometrically. When graphed, dependent systems appear as a single line where both equations overlap, leading to infinitely many solutions since any point on the line is a solution to both equations.
Dependent systems reflect scenarios where the given equations describe the same constraint or relationship between variables, rendering one of them redundant.

Infinite Solutions

When dealing with dependent systems, they always result in infinite solutions. This is because dependent equations represent the same line on a graph. Thus, rather than intersecting at a single point (as independent systems with unique solutions would), they coincide completely.
For the system \( x - 5y = 2 \) and \( 3x - 15y = 6 \), every (\(x, y\)) pair that satisfies one equation will satisfy the other, as they are different expressions of the same line. This results in infinitely many solutions, because there is not just one unique point but an entire line of points that solve both equations simultaneously.

• This concept is crucial as it signifies that there are multiple ways to fulfill the conditions the equations describe, reflecting potential flexibility in real-world situations.

Equation Multiplication

Equation multiplication is a simple yet powerful technique used to reveal relationships between equations. By multiplying an equation by a constant, we do not change its set of solutions, only its appearance.
In our exercise, multiplying the first equation \( x - 5y = 2 \) by 3 transforms it into \( 3x - 15y = 6 \), making it identical to the second equation. This transformation helps identify dependent systems by showing that the equations are not unique, but rather parallel representations of the same relationship.

• This technique is essential to recognize and understand, particularly in solving systems where equations might initially seem different but actually convey the same information.

General Solution Expression

Expressing a general solution involves finding a formula that represents all possible solutions of a system with infinite solutions. Since the equations are dependent, we can express any variable in terms of one of the other variables.
For our system, isolating \( x \) in \( x - 5y = 2 \), we get \( x = 5y + 2 \). This equation expresses \( x \) entirely as a function of \( y \), allowing us to describe the entire solution set with a single statement.
This expression is very useful because:

• It succinctly encapsulates all the possible solutions.
• It can be used to easily find solutions by substituting any real number for \( y \), reflecting the infinite number of solutions.

Overall, understanding how to express such solutions is key for interpreting the results of dependent systems.