Problem 36 - Algebra and Trigonometry

Question

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{aligned}-3 x+5 y &=2 \\ 9 x-15 y &=6 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system has no solution because the equations are parallel and inconsistent.

Step 1: Write Down the Equations

We have the system of equations: \(-3x + 5y = 2\) and \(9x - 15y = 6\).

Step 2: Recognize Linear Dependence

Observe that the second equation, \(9x - 15y = 6\), is a multiple of the first equation, \(-3x + 5y = 2\). Specifically, it multiplies the first by -3.

Step 3: Confirm Linear Dependence

Multiply the first equation, \(-3x + 5y = 2\), by -3: \ \( -3(-3x + 5y) = -3(2) \) which simplifies to \(9x - 15y = -6\). This does not match \(9x - 15y = 6\).

Step 4: Conclude No Solution

Since the equations are parallel (proportional yet yielding different constants), there is no point of intersection, therefore no solution exists.

Key Concepts

Linear Dependence

In the field of linear algebra, linear dependence refers to a situation where one or more equations in a system can be expressed as a scalar multiple of another equation. This means the equations do not bring new information to the system.
In this problem, we have the equations:

• \(-3x + 5y = 2\)
• \(9x - 15y = 6\)

When we examine these equations, we can notice something peculiar. If you multiply the first equation by -3, it resembles the form of the second equation, except for a startling difference in the constant term:

• Multiply: \(-3(-3x + 5y) = -3(2)\) gives \(9x - 15y = -6\).

The coefficients of \(-3\) and \(9\); \(5\) and \(-15\) confirm linear dependence, yet the constants \(-6\) and \(-6\). The different constant leads to interesting conclusions, showing that while the equations are related, they do not align to form a consistent system.

No Solution

In a system of linear equations, having no solution implies that the lines represented by these equations do not intersect at any point in the plane. A key indicator of this is parallel lines that have different constant terms.
In the above system, \(-3x + 5y = 2\) and \(9x -15y = 6\), the equations do not have a consistent right-hand side after checking the linear dependence. The constants \(2\) and \(6\) are not multiples of one another, leading us to conclude that the lines have different intercepts.
These different intercepts mean there is no point \((x, y)\) that satisfies both equations simultaneously. Therefore, the system of equations lacks a common solution, depicted graphically as parallel lines with a gap between them.

Parallel Lines

Parallel lines in a coordinate plane have the same slope, making them impossible to intersect. This property is crucial when analyzing systems of equations for solutions.
For equations to be parallel, their coefficients of \(x\) and \(y\) must be proportional. In our equations, both entries share the proportional relationship:

• The coefficient ratio of \(x\) to \(x\) is \(-3/9 = 1/3\).
• Similarly, the coefficient ratio of \(y\) to \(y\) is \(5/15 = 1/3\).

Notice how despite having proportional coefficients, the parallel lines arise due to the discrepancy in final constants. Because their constants, \(2\) and \(6\), are not proportional, the lines will never meet, affirming these lines are indeed parallel, underlying the reason no solutions exist.