Problem 102 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for the indicated variable. Assume all denominators are nonzero. $$a^{2}+b^{2}=c^{2}, \quad \text { for } b$$

Step-by-Step Solution

Answer

b = \begin{cases} -\begin{sqrt}c^2 - a^2\begin{endsqrt} \begin{or} b = \begin{sqrt}c^2 - a^2\begin{endsqrt} \begin{endcases}

Step 1: Isolate the term involving the variable

Re-write the equation to isolate the term involving the variable you are solving for. To isolate b^2, we need to subtract a^2 from both sides of the equation: \[ a^2 + b^2 = c^2 \]Subtract a^2 from both sides:\[ b^2 = c^2 - a^2 \]

Step 2: Solve for the variable

To solve for b, take the square root of both sides of the equation. Remember to consider both positive and negative roots when taking the square root:\[ b = \begin{cases} -\begin{sqrt}c^2 - a^2\begin{endsqrt} \begin{or} b = \begin{sqrt}c^2 - a^2\begin{endsqrt} \begin{endcases} \]

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the foundation of solving equations in algebra. It involves rearranging and simplifying expressions to isolate the variable of interest. In our exercise, we started with the equation of Pythagoras' theorem: \[ a^2 + b^2 = c^2 \]To solve for the variable \(b\), our first step was to move all terms not containing \(b\) to the other side of the equation. We did this by subtracting \(a^2\) from both sides, resulting in \[ b^2 = c^2 - a^2 \]This step is critical because it sets us up to isolate and solve for \(b\) by simplifying the equation. This type of rearrangement is called algebraic manipulation.

Isolating Variables

Isolating variables is crucial when solving for an unknown in an equation. Once we've manipulated the equation, our goal is to get the variable we're solving for (\(b\) in this case) alone on one side of the equation. In the equation \[ b^2 = c^2 - a^2 \], we have effectively isolated \(b^2\) on one side. This makes it easier to find the value of \(b\) by taking the relevant steps, such as square roots. Isolating variables often requires basic operations like addition, subtraction, multiplication, and division. Each step simplifies the equation further, making it easier to solve.

Square Roots

The concept of square roots is essential when dealing with equations involving quadratic expressions. A square root 'undoes' the squaring of a number. To find \(b\) in the equation \[ b^2 = c^2 - a^2 \], we take the square root of both sides. This gives us \[ b = \pm \sqrt{c^2 - a^2} \]Remember, taking the square root of a number yields two solutions: one positive and one negative. This is because both \( (\sqrt{x})^2 = x \) and \((-\sqrt{x})^2 = x\). Hence, \(b\) can be either \( \sqrt{c^2 - a^2} \) or \( -\sqrt{c^2 - a^2} \). By considering both solutions, we ensure the comprehensive answer, which is a key part of solving quadratic equations involving square roots.