Problem 46 - Algebra 2 and Trigonometry

Question

In \(43-46,\) solve each equation for the variable. $$ y^{2}-169=0 $$

Step-by-Step Solution

Answer

The solutions are \( y = 13 \) and \( y = -13 \).

Step 1: Rewrite Equation

Start with the original equation: \[ y^2 - 169 = 0 \] Move 169 to the other side to isolate the term with \( y^2 \): \[ y^2 = 169 \]

Step 2: Apply Square Root

Take the square root of both sides to solve for \( y \). Remember to consider both the positive and negative roots: \[ y = \sqrt{169} \]

Step 3: Simplify the Square Root

Calculate the square root of 169. Since 169 is a perfect square, \sqrt{169} = 13. Thus, the two roots are: \[ y = 13 \quad \text{and} \quad y = -13 \]

Key Concepts

Solving Equations

To solve equations, the goal is to find the value of the variable that makes the equation true. Here, we have an equation in the format \( y^2 - 169 = 0 \). The first step involves isolating the term with the variable. This is commonly done by performing inverse operations.

In our equation, we start by adding 169 to both sides, resulting in \( y^2 = 169 \). This simplification keeps the equation balanced while making it easier to solve for \( y \). Always remember, what you do to one side of the equation, you must do to the other to maintain equality.

Solving equations often requires a combination of arithmetic operations and algebraic manipulations. Here, isolating the variable term is a crucial first step. After isolation, you can shift focus to finding the actual values of the variable.

Square Root Method

The square root method is a useful technique for solving quadratic equations, particularly when the equation is in the form of \( y^2 = c \), where \( c \) is a constant. In our equation \( y^2 = 169 \), we can employ this method to find \( y \).

Our task is to find what \( y \) squared equals to 169. To achieve this, take the square root of both sides. It's crucial to consider both the positive and negative square roots because squaring either a negative or a positive number results in a positive value.

• \( y = \sqrt{169} \)
• \( y = -\sqrt{169} \)

By applying the square root method, we are directly solving for the variable, understanding that a quadratic equation often has two solutions.

Perfect Squares

A perfect square is a number that is the square of an integer. Understanding perfect squares is important because they show up often during the process of solving quadratic equations. In our scenario, the number 169 is identified as a perfect square.

To verify, we calculate \( \sqrt{169} \), which equates to 13, a whole number. This confirms that 13 is the integer whose square results in 169. Knowing the concept of perfect squares helps one recognize patterns in quadratic equations and simplifies the process of taking square roots.

For the equation \( y^2 = 169 \), realizing 169 is a perfect square instantly informs us of the solutions \( y = 13 \) and \( y = -13 \). Recognizing perfect squares thus facilitates quicker and more informed equation-solving strategies.