Problem 78 - Elementary Algebra

Question

For the following problems, solve the equations. $$ \sqrt{2 a+1}-10=-3 $$

Step-by-Step Solution

Answer

Question: Solve the equation \(\sqrt{2a+1}-10=-3\) for the variable 'a'. Answer: The solution to the equation is \(a=24\).

Step 1: Isolate the square root term

Add 10 to both sides of the equation to isolate the square root term. $$ \sqrt{2 a+1}-10+10=-3+10\\ \sqrt{2 a+1}=7 $$

Step 2: Square both sides

Square both sides of the equation to eliminate the square root. Remember to square both the left and right side of the equation. $$ (\sqrt{2a+1})^2 = 7^2 \\ 2a+1 = 49 $$

Step 3: Solve for 'a'

Subtract 1 from both sides of the equation, then divide by 2 to solve for 'a'. $$ 2a+1-1 = 49-1 \\ 2a = 48 \\ a = \frac{48}{2} \\ a = 24 $$ The solution to the equation is \(a=24\).

Key Concepts

Square Root Equations

When solving square root equations, the main goal is to eliminate the square root to solve for the variable within it. Square root equations generally involve an expression under the square root on one side of the equation, and some other terms or numbers on the other side. This is often seen as a square root equating to a number or another expression that must be dealt with to find the variable's value.
For example, the equation \( \sqrt{2a+1} - 10 = -3 \) contains a square root that must be isolated before solving the equation.

• Isolating the Square Root: Start by isolating the square root term on one side of the equation. This often involves adding or subtracting terms on both sides of the equation.
• Squaring Both Sides: Once the square root is isolated, square both sides of the equation to remove the square root. This step changes the equation into a simpler, quadratic type that can be solved with basic algebraic methods.

Keep in mind, solutions derived after squaring both sides must be checked in the original equation to ensure they are valid, as squaring can sometimes introduce extraneous solutions that don’t satisfy the original equation.

Isolation of Variables

The process of isolating variables is a fundamental skill in solving equations. It involves manipulating the equation so that the variable you're solving for is alone on one side, and all other terms are on the other side. This is done through a series of reversible arithmetic operations.
In the example equation, \( \sqrt{2a+1} = 7 \), your goal is to isolate \( a \) by performing operations that gradually simplify the equation.

• Reversing Operations: Start from operations that reverse the effect of the term being isolated. For square root equations, you'll add or subtract constants first (as seen by adding 10 in the original example).
• Utilizing Inverse Operations: Use operations such as division or multiplication based on what is present in the equation, like dividing by 2 after squaring and subtracting all constants in our example.

This step-by-step breakdown allows you to clearly see the path from the original equation to finding the solution for the variable.

Algebraic Manipulation

Algebraic manipulation involves the use of standard algebraic operations to rearrange and simplify equations. Mastering these techniques is crucial for solving any kind of algebraic equation, including square root equations.
In the provided example, once you have simplified the square root terms and squared both sides of the equation \( \sqrt{2a+1} = 7 \), the next step involves basic algebraic manipulation.

• Simplifying Equations: Eliminate additional numbers by using subtraction or addition, such as subtracting 1 from both sides in the final steps.
• Solving for the Variable: Combine like terms and perform division or multiplication to isolate the variable completely, ensuring the simplest form of the equation with respect to the variable.

Effective algebraic manipulation requires careful tracking of equalities to avoid mistakes and ensure that the solution is both valid and simplified to its most straightforward form.