Problem 7 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{a}-2=7 $$

Step-by-Step Solution

Answer

Answer: The value of \(a\) is 81.

Step 1: Isolate the square root

To isolate the square root, we'll add 2 to both sides of the equation:$$ \sqrt{a}-2+2=7+2 $$

Step 2: Simplify the equation

Simplify the equation to get:$$ \sqrt{a} = 9 $$

Step 3: Square both sides of the equation

To eliminate the square root, we'll square both sides of the equation:$$ (\sqrt{a})^2 = 9^2 $$

Step 4: Solve for \(a\)

After squaring both sides of the equation, we can find the value of \(a\):$$ a = 81 $$So, the value of \(a\) is 81.

Key Concepts

Square Root

When we talk about square roots, it involves finding a number which, when multiplied by itself, results in the given number. In our exercise, we need to work with the square root of \(a\). The expression \(\sqrt{a}\) indicates the "square root of \(a\)," meaning a number that, when squared, equals \(a\).

The square root is essential when solving equations, particularly when it appears in an equation and we need to find the unknown variable. In this specific problem, \(\sqrt{a} = 9\), we understand that 9 is the result of the square root of \(a\). To remove the square root from an equation, squaring both sides is a crucial step, as indicated later in the solution.

Recognizing the square root in equations will help you approach problems methodically. The key is not only understanding the square root symbol but also knowing how to manipulate it in various mathematical situations.

Isolation Technique

The isolation technique is a fundamental method used in solving equations, particularly when you need to focus on getting the variable alone on one side of the equation. At the beginning of our exercise, we dealt with \(\sqrt{a} - 2 = 7\). Our goal was to isolate \(\sqrt{a}\) by itself on one side.

To achieve this, we perform operations that systematically eliminate other terms. In this case, we add 2 to both sides of the equation, which cancels out the \(-2\) and moves \(\sqrt{a}\) to stand alone.

• Start by reversing operations occurring to the variable; if something is subtracted, add it back.
• Ensure you perform the same operation on both sides of the equal sign to maintain the equation's balance.

Successfully isolating an expression forms the basis for tackling more complex steps, such as squaring to remove the square root.

Equation Simplification

Equation simplification is the process of making an equation easier to handle. This involves combining like terms, reducing fractions, or performing basic arithmetic operations. In our example, once the square root is isolated, you simplify to obtain \(\sqrt{a} = 9\).

Simplification is crucial because it makes equations more manageable, allowing us to see clearer pathways to the solution. After isolating \(\sqrt{a} = 9\), further simplification was straightforward since it revealed that the next step involved treating \(\sqrt{a}\) as a number rather than an operation to apply.

• Look for parts of the equation that can be combined or reduced.
• Use standard arithmetic operations to simplify expressions.

Simplification can sometimes seem simple, but it provides a bridge to more intricate solutions, such as solving for unknowns using other operations.