Problem 16 - Calculus for Biology and Medicine

Question

Reduce the system of linear equations to upper triangular form and solve. $$ \begin{array}{r} x-2 y=2 \\ 4 y-2 x=-4 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions given by \(x = 2 + 2y\).

Step 1: Write the system of equations

The system of equations is given as: 1) \(x - 2y = 2\) and 2) \(-2x + 4y = -4\). Our goal is to eliminate one variable to create an upper triangular form.

Step 2: Eliminate x from the second equation

To eliminate \(x\) from the second equation, we can multiply the first equation by 2 and add it to the second equation. Start by multiplying the first equation by 2: \(2(x - 2y) = 4\) which simplifies to \(2x - 4y = 4\). Now add this result to the second equation: \(-2x + 4y + (2x - 4y) = -4 + 4\). This simplifies to \(0 = 0\).

Step 3: Interpret the results

The result \(0 = 0\) suggests that the two equations are not independent and the system has infinitely many solutions, but they lie along a line described by the relationship in either equation.

Step 4: Solve for one variable in terms of the other

We already have the relationship \(x - 2y = 2\). Solving for \(x\) in terms of \(y\), we get \(x = 2 + 2y\). This equation can represent the set of all solutions.

Key Concepts

Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations that involve the same set of variables. In the exercise, we have two equations involving variables \(x\) and \(y\). A system can have one solution, no solutions, or infinitely many solutions depending on how the lines represented by these equations interact with each other.

• **One solution**: If the lines intersect at a single point, the system has a unique solution.
• **No solution**: If the lines are parallel and distinct, they will never intersect.
• **Infinitely many solutions**: If the lines are coincident, meaning they lie on top of each other, then every point on the line is a solution.

In this case, as we manipulate the equations and reduce them, we'll discover that these lines are equivalent and therefore coincide.

Upper Triangular Form

The goal of transforming a system of equations into upper triangular form is to simplify it for easier solving. An upper triangular form is when all the coefficients below the main diagonal (from the top left to the bottom right) are zeros. This simplifies the system significantly and generally makes it easier to perform back substitution for finding solutions.

In our exercise, when we manipulate the equations to eliminate \(x\) from the second equation, we create a scenario where the resulting form shows that the equations are dependent. This realisation arises as we obtain the statement \(0=0\), emphasizing the nature of these dependent equations.

Infinitely Many Solutions

Discovering that a system has infinitely many solutions can at first seem perplexing, but it simply means that there are numerous solutions that satisfy both equations. In the context of linear algebra, this typically means that rather than intersecting at a single point, the lines mentioned in the system of equations are coincident.

In our example, the simplification leads to the identity \(0 = 0\), confirming the same line is represented in two different ways. By solving for one variable in terms of the other, such as solving \(x = 2 + 2y\), we find a parametric representation of the solution set, showcasing the infinite possibilities of \(y\) and corresponding \(x\) values that satisfy both equations. This highlights the relationship between the variables along the line they describe.