Problem 15 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt{a-3}=1 $$

Step-by-Step Solution

Answer

The solution is \( a = 4 \).

Step 1: Understand the Equation

The given equation is \( \sqrt{a-3} = 1 \). This equation states that the square root of \( a-3 \) is equal to 1.

Step 2: Eliminate the Square Root

To solve for \( a \), we need to eliminate the square root. We do this by squaring both sides of the equation: \( (\sqrt{a-3})^2 = 1^2 \). This simplifies to \( a-3 = 1 \).

Step 3: Solve for 'a'

Now, we solve for \( a \) by adding 3 to both sides of the equation: \( a - 3 + 3 = 1 + 3 \). This simplifies to \( a = 4 \).

Step 4: Verify the Solution

To ensure our solution is correct, we can substitute \( a = 4 \) back into the original equation: \( \sqrt{4 - 3} = 1 \). Simplifying, we have \( \sqrt{1} = 1 \), which is true. Thus, \( a = 4 \) is a valid solution.

Key Concepts

Eliminating Square Roots in Equations

The process of solving equations with square roots often involves first eliminating the square root to simplify the expression. This is key to solving the equation effectively.

• Identify the equation and locate the square root that needs to be removed. For instance, in the given equation, \( \sqrt{a-3} = 1 \), the square root is primarily on the left side.
• Square both sides of the equation to eliminate the square root. This means you will perform the operation \((\sqrt{a-3})^2 = 1^2\), effectively clearing the square root and leaving you with \(a-3 = 1\).

Squaring both sides is a critical step, as it allows us to work with simpler, non-radical expressions. It’s important to ensure each side balances equally, maintaining the equation’s integrity.

Verification of Solutions

Verifying your solution is an essential step in solving mathematical equations. This ensures that the solution obtained actually satisfies the original problem.

• After obtaining the solution, substitute it back into the original equation. For example, we calculated that \(a = 4\).
• Use this value in the initial equation \(\sqrt{a-3} = 1\). Substitute \(a\) to get \(\sqrt{4-3} = 1\), which simplifies to \(\sqrt{1} = 1\).
• Check the result to confirm equivalence on both sides of the equation. If both sides are equal, like in this case where \(1 = 1\), then \(a = 4\) is indeed a valid solution.

Verification is crucial since it not only confirms the solution but also builds your confidence in solving similar problems.

Simplifying Equations to Solve Easily

Simplifying equations is integral to making them easier and more approachable to solve. This often involves reducing equations to the simplest forms.

• After eliminating square roots, focus on reducing the equation into an easily solvable form. In our example, the equation was simplified to \(a - 3 = 1\).
• Work with numbers by performing arithmetic operations that will isolate the variable. Add or subtract terms as needed. Here, adding 3 to both sides gives \(a = 4\).

Making the equation simpler saves time and reduces the likelihood of errors. It allows for straightforward operations and helps you to clearly understand each step you take towards the solution. As a result, you're more likely to get accurate solutions efficiently.