Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

You graphically approximate the solution of the equation \(\frac{x}{x-1}-\frac{99}{100}=0\) to be \(x=-99.1 .\) Substituting this value for \(x\) produces \(\frac{-99.1}{-99.1-1}-\frac{99}{100}=0.00000999=9.99 \times 10^{-6}\) Is -99.1 a good approximation of the solution? Write a short paragraph explaining why or why not.

Step-by-Step Solution

Answer

Yes, \(x=-99.1\) is a good approximation of the solution to the problem. This is because when substituted into the equation, it results in a value extremely close to zero, indicating the equation is nearly balanced. However, an ideal solution would balance the equation exactly, resulting in zero.

Step 1: Understanding the Original Equation

The original equation given is \(\frac{x}{x-1}-\frac{99}{100}=0\). We can manipulate this equation to a form which can be easily substituted by the proposed solution. From the original equation, we observe that it can be simplified to \(x = 99x - 99\). From there, we can then simplify it further to the form \(x - 99x = 99\)

Step 2: Substituting the proposed solution

The proposed solution to the equation is \(x=-99.1\). Let us substitute this solution into the simplified equation, so it becomes \(-99.1 - 99 \times -99.1 = 99\). When this is solved, we end up with \(9.99 \times 10^{-6}\).

Step 3: Evaluating the Approximation

The result after substituting the proposed solution into the equation results in a very small number (almost zero). This means that the equation is close to being balanced, so it could be inferred that \(x=-99.1\) can be considered as a good approximation of the solution to the problem. However, an ideal solution would have resulted in an equation that is exactly balanced resulting in 0.

Key Concepts

Graphical Solution Approximation

When tackling algebraic equations, students often find graphical solution approximation to be a powerful tool. This method involves plotting the function represented by the equation on a graph and visually identifying where it crosses the x-axis, which corresponds to the solution of the equation.

For the given equation \(\frac{x}{x-1}-\frac{99}{100}=0\), you would plot the left side of the equation as a function of \(x\) and look for the x-value where the function's y-value is zero. Using this method, it's found that \(x \approx -99.1\). While this is helpful for a general understanding, it may not be exact. Precision in the graphical method depends on scale and resolution of the graph, and it's always beneficial to confirm by using algebraic techniques. Therefore, even if \(x=-99.1\) appears to be a root on the graph, it may not be the exact solution when plugged back into the equation.

Substitution Method

The substitution method is a fundamental algebraic technique used to determine the exact solution of an equation. After simplifying the equation as in Step 1 from the original exercise, we substitute the approximate solution back into the simplified equation to see how close we are to zero. In this case, substituting \(x = -99.1\) may lead to a small nonzero value.

However, a perfectly accurate solution would lead to a zero, confirming that the equation is in balance. The closer the result is to zero after the substitution, the better the approximation. It's worth noting that in algebra, an exact solution is always preferred to an approximation, which is why we use substitution as a method to check the accuracy of our graphical estimates.

Evaluating Algebraic Expressions

Evaluating algebraic expressions is a skill that involves substituting numbers for variables and performing the arithmetic operations as indicated in the expression. When we assess the approximation for \(x\) by substituting \(x = -99.1\) into the equation, we are performing an evaluation.

If the substituted value of \(x\) results in an expression that simplifies very close to zero, it indicates we may have a good approximation. On the contrary, a larger number suggests a less accurate approximation. The evaluation in this context is crucial because it verifies the validity of the graphical solution approximation. It is essentially what answers the question of whether \(x=-99.1\) is a good approximation of the solution.

Simplifying Algebraic Equations

Simplifying algebraic equations is a process which involves reducing an equation to its simplest form, making it easier to solve or analyze. This usually involves combining like terms, eliminating fractions, and isolating the variable. In our example, simplifying the original equation allows us to clearly see the relationship between \(x\) and the constants involved.

By simplifying the equation, we enhance our ability to substitute values and evaluate how close we come to an exact solution. The goal in simplification is to make the equation as easy to work with as possible, thus preventing errors in the subsequent steps of problem-solving and getting us closer to understanding the true solution.