Problem 84 - Algebra A Combined Function

Question

In your own words, explain how to check a solution of an equation.

Step-by-Step Solution

Answer

Verify by substituting the solution into the original equation and checking for equality on both sides.

Step 1: Understand the Equation

Begin by carefully reading the given equation. Identify all variables and constants involved. Make sure you understand the relationship that the equation is expressing.

Step 2: Solve the Equation

Solve the equation for the unknown variable(s). To do this, perform algebraic manipulations such as adding, subtracting, multiplying, dividing, or using more advanced techniques like factoring or using the quadratic formula, depending on the type of equation.

Step 3: Substitute the Solution

Take the solution obtained in the previous step and substitute it back into the original equation. Replace the variable(s) with the value(s) found.

Step 4: Simplify Both Sides

Calculate or simplify both sides of the equation after substituting the solution. Take care to perform any necessary arithmetic operations or simplify expressions on both sides.

Step 5: Verify Equality

Check if both sides of the equation are equal. If the left-hand side (LHS) equals the right-hand side (RHS) after simplification, the solution is correct.

Key Concepts

Algebraic Manipulations

Algebraic manipulations are the tools we use to solve equations. The goal is to isolate the variable we want to find. There are several methods to do this, which involve basic operations like addition, subtraction, multiplication, and division. However, equations aren't always straightforward. We often need to use more advanced techniques. For example:

• Factoring: Breaking down a complex expression into simpler parts.
• Distributing: Expanding expressions to simplify or combine them with others.

Using these methods strategically allows us to simplify the problem step by step. Remember, each manipulation should bring us closer to finding the variable's value. Mastering these techniques is crucial, as they are the foundation for solving more complex problems involving multiple variables or higher-degree equations.

Substituting Solutions

Once we have a potential solution, substituting involves testing it in the original equation to ensure it works. This step is vital because it confirms our earlier manipulations. Here’s how it works:

• Take the variable's value you found and plug it back into the equation wherever the variable appears.
• Replace every instance of the variable with the solution obtained.

Substitution helps reconfirm that the solution derived satisfies the original mathematical statement. By doing so, you not only check your work but also build confidence in your problem-solving process.

Verifying Solutions

After substituting, we must verify that our solution is correct. Verification is the final checkpoint in equation solving, where you compare both sides of the equation. In practice:

• Simplify both sides as needed after substitution.
• Check if the left-hand side (LHS) equals the right-hand side (RHS).

If they do, congratulations! You've found a valid solution. If not, revisit your previous steps. Verification ensures that the solution not only mathematically works but also remains consistent with the original problem.

Quadratic Formula

One of the fundamental tools in solving second-degree equations is the quadratic formula. It comes in handy when factoring is complex or impractical. The formula is:\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]Here's how to use it:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c = 0 \).
• Substitute these values into the formula.
• Calculate the solutions by performing the arithmetic operations.

The quadratic formula gives precise solutions, ensuring no valid root is overlooked. Understanding this formula expands your problem-solving toolkit, allowing you to tackle a wide range of quadratic equations with confidence.