Question

Use Jacobi's method to find the solution of $$ \begin{aligned} -2 x+y &=8.5 \\ 3 x+4 y &=1 \end{aligned} $$ Take \(x_{0}=-2, y_{0}=2\) and perform five iterations.

Short answer

\[-2x + y = 8.5\] \[3x + 4y = 1\] Answer: After five iterations of Jacobi's method, the approximate solution is \(x \approx 6.328125\) and \(y \approx -4.16015625\).

Step-by-step solution

Rewrite the equations with one variable isolated

First, let's rewrite both equations to isolate \(x\) and \(y\) on the left side of the equals sign. From Equation 1: $$ -2x + y = 8.5$$ isolate \(x\) as: $$ x = \frac{1}{2}(8.5 - y)$$ From Equation 2: $$ 3x + 4y = 1$$ isolate \(y\) as: $$ y = \frac{1}{4}(1 - 3x)$$

Initialize variables and iterations

Now, let's initialize the variables \(x\) and \(y\) with the given values: \(x_0 = -2\) and \(y_0 = 2\). We will perform five iterations, so we need a loop with five steps.

Perform five iterations using Jacobi's method

Perform the following steps five times: 1. For the current values of \(x\) and \(y\), update \(x\) using the first equation: \( x = \frac{1}{2}(8.5 - y)\) 2. For the current values of \(x\) and \(y\), update \(y\) using the second equation: \( y = \frac{1}{4}(1 - 3x)\) 3. Repeat steps 1 and 2 for a total of five iterations.

Jacquard's method iterations

Let's perform the iterations: Iteration 1: \(x_1 = \frac{1}{2}(8.5 - 2) = 3.25\) \(y_1 = \frac{1}{4}(1 - 3(-2)) = 1.75\) Iteration 2: \(x_2 = \frac{1}{2}(8.5 - 1.75) = 3.375\) \(y_2 = \frac{1}{4}(1 - 3(3.25)) = -2.375\) Iteration 3: \(x_3 = \frac{1}{2}(8.5 - (-2.375)) = 5.4375\) \(y_3 = \frac{1}{4}(1 - 3(3.375)) = -2.78125\) Iteration 4: \(x_4 = \frac{1}{2}(8.5 - (-2.78125)) = 5.640625\) \(y_4 = \frac{1}{4}(1 - 3(5.4375)) = -4.15625\) Iteration 5: \(x_5 = \frac{1}{2}(8.5 - (-4.15625)) = 6.328125\) \(y_5 = \frac{1}{4}(1 - 3(5.640625)) = -4.16015625\)

Final solution

After five iterations of Jacobi's method, we have the following approximated solution: $$ x \approx 6.328125$$ $$ y \approx -4.16015625$$

Key concepts

System of Linear Equations

In the world of algebra, a system of linear equations consists of two or more equations that are solved simultaneously. Each equation within the system typically represents a line, and the solution to the system corresponds to the point(s) where these lines intersect.

For example, in the given exercise, we are dealing with a system of two linear equations with two variables, which can be visualized as two straight lines on a graph. The exercise tasks us with solving the system using Jacobi's method, an iterative numerical technique, which is particularly useful when the system is large or when algebraic methods are not convenient.

One key to understanding these systems is recognizing that the variables in each equation are dependent on each other, and a change in one affects the value of the other. Thus, solving such systems requires methods that consider the entire system simultaneously, unlike solving a single equation where the focus is on isolating and solving for one variable at a time.

Iterative Methods for Solving Equations

Iterative methods, such as Jacobi's method demonstrated in the exercise, are an essential part of numerical mathematics. They are designed to refine an initial guess into an increasingly accurate solution by repeating a set of procedures.

These methods are particularly useful when an exact solution is difficult to find or when we have to deal with large scale problems where direct methods become computationally expensive. Unlike direct methods, iterative techniques do not aim to produce a solution in a finite number of steps. Instead, they work by successively approximating a more accurate solution, iterating the process until the results converge within a desired level of accuracy.

During the iteration processes for the Jacobi's method, new estimates for the variables are generated using only the values from the previous iteration, avoiding direct dependency on the current step, thus preventing simultaneous equation solving and allowing for parallel computations.

Numerical Methods in Engineering Mathematics

Numerical methods are the backbone of engineering mathematics when it comes to solving real-world problems. These are a broad category of algorithms used to find approximate solutions to complex mathematical problems. Engineering applications often result in equations that either cannot be solved analytically or are too time-consuming to solve symbolically.

Techniques like Jacobi's method help engineers and scientists to model and analyze systems ranging from structures to dynamic fluids, where exact solutions are either unobtainable or impractical. Being iterative in nature, these methods allow for the evaluation of large systems like climate models or stress analysis simulations with remarkable efficiency.

Understanding the theory and application of these numerical methods is crucial for producing results that are not only fast but also accurate. For students stepping into the field of engineering, a firm grasp of these methods is critical, as they will be the tools used to translate complex theoretical models into practical solutions that can be applied in the real world.