Problem 5 - Calculus for Biology and Medicine

Question

Determine \(c\) such that $$ \begin{array}{l} 2 x-3 y=5 \\ 4 x-6 y=c \end{array} $$ has (a) infinitely many solutions and (b) no solutions. (c) Is it possible to choose a number for \(c\) so that the system has exactly one solution? Explain your answer.

Step-by-Step Solution

Answer

(a) Infinitely many solutions: \( c = 10 \). (b) No solutions: \( c \neq 10 \). (c) No, it's not possible for exactly one solution.

Step 1: Understanding the System of Equations

We are given a system of two equations: \(2x - 3y = 5\) and \(4x - 6y = c\). The first task is to understand when the system can have infinitely many solutions, no solutions, or exactly one solution.

Step 2: Setting Conditions for Infinitely Many Solutions

For the system of equations to have infinitely many solutions, both equations must be consistent and dependent, meaning they are essentially the same line. The second equation should be a multiple of the first. Here, \(4x - 6y\) is a multiple of \(2x - 3y\). Set the constants such that \(5\) and \(c\) hold the same ratio as their coefficients: \( \frac{4}{2} = \frac{c}{5} \). Solving \( \frac{4}{2} = \frac{c}{5} \) gives \( c = 10 \).

Step 3: Setting Conditions for No Solutions

The system will have no solutions if the equations represent parallel lines, which means they have the same slope but different intercepts. This requires the left-hand sides of the equations to remain proportional while the constants on the right-hand do not maintain the same ratio: \( \frac{4}{2} eq \frac{c}{5} \). Hence, any value of \(c\) other than \(10\) will create no solutions, such as \( c = 9 \).

Step 4: Exploring the Possibility of Exactly One Solution

For a system of two linear equations to have exactly one solution, the lines must intersect at a single point, implying they are not parallel and not coincident. However, here \(4x - 6y\) is exactly twice \(2x - 3y\) in terms of coefficients, meaning the lines are always parallel (or coincident). Consequently, it is impossible to choose any \(c\) resulting in exactly one solution.

Key Concepts

Infinitely Many Solutions in Linear Systems

In linear algebra, a system of equations is said to have infinitely many solutions when the equations represent the same line. Two lines are considered coincident when each point on one line lies on the other, resulting in countless solutions that make both equations true simultaneously.
When examining the system:

• \(2x - 3y = 5\)
• \(4x - 6y = c\)

it becomes clear that the second equation is simply a scaled version of the first. To guarantee infinitely many solutions, the constants on the right side of each equation must maintain proportionality with the coefficients of operations on the variables.
Considering \[\frac{4}{2} = \frac{c}{5}\]we find that when \[c = 10\]the two equations describe the same line. Thus, with this value of \(c\), there are infinitely many solutions.

Understanding No Solution in Linear Systems

Linear systems have no solution when the lines they represent are parallel but not the same. Parallel lines maintain the same slope, which is why their coefficients are proportional, but they have different intercepts, leading to no points of intersection.
Let's consider the system:

• \(2x - 3y = 5\)
• \(4x - 6y = c\)

By examining these equations, we recognize they can appear parallel if their left-hand sides are multiples. When \[\frac{4}{2} eq \frac{c}{5}\]this indicates parallel lines not sharing every point, having different intercepts. In practical terms, any value of \(c\) other than 10, such as \(c = 9\), will yield a system with no solutions.

Role of Linear Algebra in Solving Systems

Linear algebra provides methods and concepts for solving systems of equations efficiently. One essential element involves recognizing the geometrical interpretation of equations as lines or planes. When faced with a two-equation system, it’s crucial to examine their coefficients and constants to determine the relational characteristics.
These characteristics include:

• Lines having the same slope but differing intercepts are parallel, indicative of no solutions.
• Identical lines across equations with maintained proportional constants indicate infinitely many solutions.
• Different slopes between occurrences would generally imply a single intersection point, or one solution, however, this particular system cannot achieve that given their structure.

Linear algebra allows us to systematically solve and interpret these systems, providing insights into their nature and the type of solutions possible.