Problem 12 - Algebra 2

Question

Solve each equation. $$ \sqrt{y}-7=0 $$

Step-by-Step Solution

Answer

The value of \( y \) is 49.

Step 1: Isolate the Square Root

Add 7 to both sides of the equation to isolate the square root. The equation becomes \( \sqrt{y} = 7 \).

Step 2: Remove the Square Root

Square both sides of the equation to remove the square root. This gives \( y = 7^2 \).

Step 3: Simplify the Equation

Calculate the square of 7 to simplify the equation. \( 7^2 = 49 \), so \( y = 49 \).

Key Concepts

Square Roots

Square roots are an essential concept in mathematics. They help us find the number that," when multiplied by itself, results in the given number. For example, the square root of 49 is 7 because \( 7 \times 7 = 49 \). In the context of equations, square roots help simplify expressions and make the problem easier to solve.
When dealing with square roots in equations, it's important to handle them carefully. You may need to manipulate the equation to isolate the square root before any further steps.
Square roots also always involve positive outcomes as they conventionally result in non-negative values. This makes them slightly different from squares themselves which can be positive or negative.

Isolating Variables

In solving equations, isolating the variable is a crucial step. It means getting the variable on one side of the equation by itself.
This process often involves using operations such as addition, subtraction, multiplication, or division. In the exercise provided, the first step was to isolate the variable under the square root by adding 7 to both sides of the equation. This resulted in the equation \( \sqrt{y} = 7 \).
Isolating variables helps clarify what operation or series of operations we need to perform in order to find the solution. It breaks down complex problems into manageable parts, making it easier to solve.

Squaring Both Sides

Squaring both sides of an equation is a common technique used to eliminate square roots. This technique involves raising both sides of the equation to the power of two.
In our current exercise, after isolating the square root \( \sqrt{y} = 7 \), we then square both sides to remove the square root. This results in \( y = 7^2 \) or \( y = 49 \).
This method is particularly useful when dealing with equations involving square roots because it simplifies the equation significantly and leads to a clear, straightforward solution.

Mathematics Education

Mathematics education emphasizes understanding these fundamental concepts. Learning to solve equations step by step enhances problem-solving skills. It empowers students to tackle more complex challenges effectively.
Emphasizing concepts like square roots, isolating variables, and techniques like squaring both sides, fosters a deeper understanding of algebra. This foundational knowledge is crucial for further studies in mathematics and sciences.

• Step-by-step learning builds confidence.
• The process is as important as the answer.
• Encourages logical reasoning and analytical thinking.

The skills gained in structured mathematics education transcend the classroom, providing a valuable toolkit for real-world problem solving.