Problem 17 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt{4 x+5}=5 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 5 \).

Step 1: Understand the Equation

The given equation is \( \sqrt{4x + 5} = 5 \). This mathematical expression involves a square root equation. Our task is to solve for \( x \).

Step 2: Remove the Square Root by Squaring

To eliminate the square root, square both sides of the equation. This gives \( (\sqrt{4x + 5})^2 = 5^2 \), which simplifies to \( 4x + 5 = 25 \).

Step 3: Isolate the Variable Term

Subtract 5 from both sides to move the constant term to the right side. This results in \( 4x = 20 \).

Step 4: Solve for x

Divide both sides by 4 to solve for \( x \). This yields \( x = \frac{20}{4} \), resulting in \( x = 5 \).

Step 5: Verify the Solution

Substitute \( x = 5 \) back into the original equation to check: \( \sqrt{4(5) + 5} = \sqrt{20 + 5} = \sqrt{25} = 5 \). The left-hand side equals the right-hand side, verifying the solution is correct.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different quantities. They often include variables, constants, and mathematical operations. In the equation \( \sqrt{4x + 5} = 5 \), we have:

• A square root, which is a type of radical expression involving the variable \( x \).
• A constant number 5 on the right side of the equation.

Understanding how to work with these components is essential for solving algebraic equations. The main goal is to determine the value of the variable that makes the equation true. This usually involves a series of logical steps that manipulate the equation into a simpler form until we find the solution. Breaking down the process into manageable segments helps to better tackle the complexity of the problem.

Effective Step-by-Step Solutions

Solving equations effectively often requires a methodical step-by-step approach. Such structured solutions make it easier to identify mistakes and ensure each transformation of the equation is valid. Let’s break down the process of solving the square root equation \( \sqrt{4x + 5} = 5 \):

• Understand the Equation: Recognize the components and the operations involved, identifying it as a square root equation.
• Eliminate the Square Root: Squaring both sides removes the square root, simplifying the equation to its core components, \( 4x + 5 = 25 \).
• Isolate the Variable: Separate the variable from constants to further simplify, achieving forms like \( 4x = 20 \).
• Solve for the Variable: Isolate \( x \) completely by performing inversely related operations, like division, to find \( x = 5 \).
• Verify the Solution: Always substitute back to ensure both sides of the equation hold true with the obtained solution, confirming correctness.

Approaching problems in this way helps with clarity and builds confidence in the accuracy of final results.

Isolating the Variable in Equations

Isolating the variable is a crucial step in solving algebraic equations, especially when dealing with operations like square roots. To isolate a variable, follow these general principles:

• **Undo Operations in Reverse Order:** Often start with functions affecting the variable and peel them off, one step at a time.
• **Keep the Equation Balanced:** Always perform the same operation on both sides of the equation to keep it true. For example, in the equation \( 4x + 5 = 25 \), subtract 5 from both sides to obtain \( 4x = 20 \).
• **Use Inverse Operations:** Reverse the effect of mathematical operations. In this case, division by 4 simplifies \( 4x = 20 \) to \( x = 5 \).

These principles apply broadly to many types of equations, from linear to more complex forms. The ability to isolate variables is fundamental in algebra as it leads directly to the solutions.