Which equation is the inverse of (x-4)^2-2/3=6y-12

To find the inverse of the given equation:

Step 1: Simplify the Original Equation

First, we start by isolating $y$ on one side of the equation. Add $12$ to both sides:

Simplify:

Note that:

Thus:

Multiply through by 3:

Divide both sides by 18:

Step 2: Solve for $x$ in Terms of $y$

Swap $x$ and $y$:

Multiply by 18:

Subtract 34 from both sides:

Divide by 3:

Step 3: Rewrite the Inverse Equation

Finally, solve for $y$:

Take the square root of both sides:

Add 4 to both sides:

Thus, the inverse function is:

This gives us both possible values for the inverse function, depending on whether the positive or negative square root is taken.

Understanding Key Concepts Used in Solving and Simplifying Algebraic Equations

In algebra, solving and simplifying equations are essential skills. These processes help you find the values of unknown variables and streamline complex expressions. Below, we explore some of the key concepts that are vital in this context.

Key Concepts

1. Variables and Constants

   • Variables: Symbols (like $x$ or $y$) representing unknown values.
   • Constants: Fixed numerical values.

2. Operations

   • Addition, subtraction, multiplication, and division.
   • Use of parentheses ($($ and $)$) for grouping terms.

3. Balancing Equations

   • Ensuring both sides of the equation are equal.
   • Whatever operation you perform on one side, must be performed on the other side too.

Techniques

1. Isolating Variables
   • Move terms involving variables to one side, constants to the other.
   • For example:

2. Combining Like Terms

   • Terms with the same variable and power can be combined.
   • Example: $$2x + 3x = 5x$$

3. Factoring

   • Factoring polynomial expressions to simplify and solve.
   • Quadratic example: $$x^2 + 5x + 6 = 0 \rightarrow (x + 2)(x + 3) = 0$$
   • Solving: $$x + 2 = 0 \rightarrow x = -2$$ $$x + 3 = 0 \rightarrow x = -3$$

4. Using the Distributive Property

   • Distributing a factor across terms inside parentheses.
   • Example: $$a(b + c) = ab + ac$$
   • Solving example:

Simplifying Expressions

1. Canceling Common Factors

   • Simplify fractions by canceling common factors in the numerator and the denominator. $$\frac{6x}{3} = 2x$$

2. Combining Fractions

   • Combine fractions by finding a common denominator: $$\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$$

3. Rationalizing Denominators

   • Eliminate radicals from denominators: $$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

Quadratic Formula

For solving quadratic equations of the form

Conclusion

Mastering these concepts and techniques is crucial for solving and simplifying algebraic equations effectively. Practice regularly to enhance your algebraic problem-solving skills.

Explanation of Taking the Square Root of Both Sides

Taking the square root of both sides of an equation is a common technique used to solve for a variable. This method is particularly useful when dealing with quadratic equations or when the variable is squared.

Basic Idea

If you have an equation where a variable is squared, such as:

To solve for $x$, you would take the square root of both sides:

Important Note on Solutions

It's crucial to remember that taking the square root of both sides introduces a $\pm$ (plus or minus) symbol, because both the positive and negative values will satisfy the original squared equation. Hence, the actual equation becomes:

Example

Consider the equation:

By taking the square root of both sides, you get:

Therefore, the solutions to the equation $x^2 = 16$ are $x = 4$ and $x = -4$.

Application Steps

1. Isolate the squared term: Make sure the term containing the squared variable is by itself on one side of the equation.
2. Take the square root of both sides: Apply the square root to both sides of the equation.
3. Simplify and solve: Simplify the resulting expression and include both positive and negative solutions.

Constraints

When taking the square root of both sides, ensure that the value inside the square root (the radicand) is non-negative if dealing with real numbers. For complex solutions, different rules apply.

Conclusion

This process is a powerful and essential tool in algebra. By understanding and correctly applying the rule of taking the square root of both sides, you can effectively solve various types of equations, ensuring to not forget the $\pm$ to capture all possible solutions.