Problem 119 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$a^{2}+b^{2}=c^{2} \quad \text { for } a$$

Step-by-Step Solution

Answer

\(a = \pm \sqrt{c^2 - b^2}\)

Step 1: Isolate the term involving the variable

Start by isolating the term that contains the variable we need to solve for. In this case, we need to isolate the \(a^2\) term. Do this by subtracting \(b^2\) from both sides of the equation: \(a^2 = c^2 - b^2\).

Step 2: Solve for the variable

Now that we have isolated \(a^2\), take the square root of both sides to solve for \(a\). Remember to consider both the positive and negative roots when taking the square root: \(a = \pm \sqrt{c^2 - b^2}\).

Key Concepts

Equation Manipulation

Equation manipulation is the process of reorganizing and adjusting equations to make it easier to solve for a specific variable or value. In the case of the Pythagorean Theorem, represented by the equation \(a^2 + b^2 = c^2\), manipulating the equation helps us isolate a specific term.

Here are some simple steps to effectively manipulate equations:

• Identify the term you need to isolate. For this exercise, it's \(a^2\).
• Determine what operations are needed to separate this term from the others. If \(b^2\) is added to \(a^2\), subtract \(b^2\) from both sides to achieve \(a^2 = c^2 - b^2\).
• Ensure the modifications are performed on both sides of the equation to maintain balance.

By mastering these basic manipulations, solving complex equations becomes much more straightforward. Equation manipulation is foundational in solving algebraic problems and is essential for success in higher math studies.

Solving for a Variable

Solving for a variable means finding the value of one specific element in an equation. This is a common task when working with equations like those derived from the Pythagorean Theorem.

Here's a simple approach to solving for a variable:

• Isolate the target term as much as possible using inverse operations. This might include addition, subtraction, multiplication, or division, as needed.
• Once isolated, apply necessary mathematical operations to solve for the variable. In this case, you'll take the square root of both sides to resolve \(a\) from \(a^2 = c^2 - b^2\).
• When using operations like square roots, always consider both potential values (positive and negative) due to the nature of squares.

Successfully solving for a variable involves clear and logical rearrangement of the equation, enabling you to pinpoint the value you're interested in. This skill is pivotal in executing various algebraic applications.

Square Roots

Square roots play a vital role in algebra, especially in topics related to equations such as the Pythagorean Theorem. Understanding square roots aids in accurately determining the lengths of sides in a right triangle, among other applications.

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square roots of 9 are 3 and -3, because both \(3^2\) and \((-3)^2\) equal 9.

When working with equations like \(a^2 = c^2 - b^2\):

• Take the square root of both sides to solve for \(a\), resulting in \(a = \pm \sqrt{c^2 - b^2}\).
• The \(\pm\) symbol indicates the dual nature of square roots, acknowledging the presence of both a positive and a negative solution.

Square roots are fundamental in many mathematical operations, and understanding their properties ensures the mastery of various problem-solving tasks in both theoretical and practical scenarios.