Problem 73 - Elementary Algebra

Question

For the following problems, solve the equations. $$ \sqrt{a-7}=6 $$

Step-by-Step Solution

Answer

Based on the step by step solution provided above, answer the following question: What is the value of 'a' in the given equation: $$\sqrt{a-7} = 6$$ Answer: The value of 'a' is 43.

Step 1: Identify the given equation

The given equation is: $$ \sqrt{a-7}=6 $$

Step 2: Square both sides of the equation

To eliminate the square root, we need to square both sides of the equation, like this: $$ (\sqrt{a-7})^2 = 6^2 $$

Step 3: Simplify the equation

When we square both sides of the equation, we get: $$ a - 7 = 36 $$

Step 4: Solve for 'a'

Now, we will add 7 to both sides of the equation to isolate 'a': $$ a - 7 + 7 = 36 + 7 $$ Which simplifies to: $$ a = 43 $$

Step 5: State the solution

The solution of the given equation is: $$ a = 43 $$

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They help us find a number, which when multiplied by itself, gives the original number under the square root symbol. For example, the square root of 36 is 6 because 6 × 6 equals 36. In mathematics, it is denoted using the square root symbol \(\sqrt{}\).

In solving equations, square roots often appear on one side of the equation. The goal is to "undo" the square root so we can isolate the variable. This is typically done by squaring both sides since squaring and taking a square root are inverse operations. Remember:

• Square rooting undoes squaring and vice-versa.
• When you square both sides of an equation, make sure that every part of each side is squared.

Understanding square roots and how to manipulate them is key to solving many algebraic equations.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations, making it easier to solve them. Let's consider it in the context of solving square root equations.

Once we eliminate the square root in our example \(\sqrt{a-7} = 6\), by squaring both sides, we derived a simpler equation: \(a - 7 = 36\). This is basic algebraic manipulation as follows:

• You relocate terms to isolate the variable.
• In the equation \(a - 7 = 36\), adding 7 to both sides helps us isolate \(a\).
• Each operation must be performed uniformly on both sides of the equation.

These steps are crucial to solving equations. By understanding algebraic rules and the properties of equality, you're equipped to solve a wide range of mathematical problems effectively.

Equation Solving Steps

Solving equations is a process that can be broken down into manageable steps. Each step brings you closer to finding the value of the unknown variable. Here is a systematic approach that we followed in our solution:

1. **Identify the Equation**: Start by clearly writing down the given equation: \(\sqrt{a-7} = 6\). Grasp the nature of the equation to determine the best method of solving.
2. **Eliminate the Square Root**: Since square roots confuse direct solving, square both sides to simplify this, transforming \(\sqrt{a-7} = 6\) into \(a - 7 = 36\).
3. **Isolate the Variable**: Use algebraic manipulation to get the variable on one side of the equation. Add, subtract, multiply, or divide both sides as needed.
4. **Solve for the Variable**: This might involve combining like terms or dividing to solve for the variable: \(a = 43\).
5. **Verify Your Solution**: Substituting the solution back into the original equation ensures that it works. Plugging \(a = 43\) into \(\sqrt{43-7} = 6\), check if it equates to 6.
By following these well-defined steps, solving equations becomes a streamlined and straightforward task. Practice enhances efficiency and accuracy.