Problem 117 - A Graphical Approach to College Algebra

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. 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: Identify the Given Equation

The given equation is the Pythagorean Theorem: \(a^2 + b^2 = c^2\). We need to solve this equation for the variable \(a\).

Step 2: Isolate the Term with Variable 'a'

Subtract \(b^2\) from both sides of the equation to isolate the \(a^2\) term. This step clears other constants away from \(a^2\):\[a^2 = c^2 - b^2\]

Step 3: Solve for 'a'

To solve for \(a\), take the square root of both sides. Remember that when you take the square root, you include both positive and negative roots, hence the \(\pm\):\[a = \pm \sqrt{c^2 - b^2}\]

Key Concepts

Solving Equations

Solving equations is all about finding the value of the unknown variable that satisfies the equation. In this case, we're working with the Pythagorean Theorem equation, which is a fundamental equation in geometry:

\(a^2 + b^2 = c^2\). This equation relates the lengths of the sides of a right triangle. To solve for a specific variable, such as \(a\), you need to perform operations that will isolate this variable on one side of the equation.

• Start by identifying the terms that need to be moved (or simplified) to achieve isolation of the variable.
• Apply the inverse operation to both sides of the equation, keeping the equation balanced.
• Each operation you perform should ensure that you are inching closer to having your variable all alone, ready to reveal its value.

By isolating \(a^2\) by subtracting \(b^2\) from both sides, you clear distractions away from the main focus of your equation. Finally, to solve \(a^2 = c^2 - b^2\) fully, you need to get \(a\), not \(a^2\). That's where the square root comes in, which happens in the next step.

Variables

Variables are symbols used in algebra to represent unknown values or quantities that can change. They are placeholders that allow equations to be both general and flexible. In the equation from the Pythagorean Theorem, \(a, b,\) and \(c\) are variables representing the lengths of the sides of a right triangle.

• \(a\) usually represents one of the legs of a right triangle.
• \(b\) is often the other leg of the triangle.
• \(c\) represents the hypotenuse, which is the side opposite the right angle and always the longest side of the triangle.

In our equation, the goal is to isolate and solve for \(a\). Each variable in the Pythagorean Theorem plays a specific role. Knowing which variable you're solving for and what it represents is key to understanding the geometric context and applying the correct operations. Always remember that the labels of variables can vary depending on the context, but their roles remain consistent.

Square Root

The square root is a special mathematical operation that finds a number which, when multiplied by itself, gives the original number. It's closely related to the concept of squaring because it reverses the squaring process. In the equation
\(a^2 = c^2 - b^2\), you're tasked with finding \(a\). This requires you to take the square root of both sides of the equation.

The square root of a number \(x\) is written as \(\sqrt{x}\). Importantly, a number has two square roots: one positive and one negative, indicated by the symbol \(\pm\). This is because both \((\sqrt{x})^2 = x\) and \((-\sqrt{x})^2 = x\).

• Apply the square root operation to \(a^2 = c^2 - b^2\) to get \(a = \pm \sqrt{c^2 - b^2}\).
• Including \(\pm\) is essential, as both roots are potential solutions unless the context specifies otherwise.

This mathematical operation helps bridge the gap from \(a^2\) to \(a\) itself, ensuring the solution is complete.