Problem 32 - College Algebra

Question

For the following exercises, solve for the variable. $$ 4 y(7-2)^{2}=-200 $$

Step-by-Step Solution

Answer

The solution is \( y = -2 \).

Step 1: Simplify the expression in parentheses

First, evaluate the expression within the parentheses, \(7 - 2\). This simplifies to \(5\).

Step 2: Simplify the exponent

Next, compute \(5^2\), which is the square of the expression inside the parentheses. \(5^2 = 25\).

Step 3: Multiply out the term

Now, multiply the inside of the parentheses by \(4y\): \(4y imes 25 = 100y\).

Step 4: Set up the equation

Replace the left-hand side of the equation with the simplified term, which becomes \(100y = -200\).

Step 5: Solve for the variable

To solve for \(y\), divide both sides of the equation by 100: \( y = \frac{-200}{100}\).

Step 6: Simplify the solution

Simplify \(\frac{-200}{100}\) to get \(y = -2\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex equations into simpler forms. In this activity, the operation starts by looking inside parentheses. The expressions inside the parentheses are calculated first due to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

• First, look inside any parentheses and perform the arithmetic operations found there.
• The step "Simplify the expression in parentheses" turned the expression \(7 - 2\) into \(5\).
• This simplification step reduces the complexity of the equation, making subsequent steps more straightforward.

Breaking down expressions into simpler components helps to make solving equations easier and reduces errors. Remember to proceed step by step and ensure each operation respects the order of operations.

Factoring

Factoring often involves breaking down a complicated expression into a product of simpler elements. Although this problem does not involve factoring in the traditional sense, understanding the multiplication of simplified expressions is essential.

• After simplifying expressions, related terms are grouped together, often involving multiplication as one of the final steps in simplification.
• In this case, after simplifying \((7-2)^2\) to \(25\), it is then factored by incorporating \(4y\). This means multiplying \(25\) by \(4y\) to obtain \(100y\).

Each part of the equation works as a piece of a puzzle, ultimately fitting together to solve for the unknown variable.

Exponents

Exponents are numbers that indicate how many times a number, called the base, is multiplied by itself. Here, they play a critical role in transforming the expression inside the parentheses.

• The initial step, \((7-2)^2\), involves an exponent applied to the result of the expression \((5\), which stands for \(5 \times 5\).
• This operation is crucial, as it transforms the expression into a product, \(25\), reducing the equation's complexity.

Working with exponents requires careful calculation to ensure that the base number is multiplied by itself the correct number of times. Mastery over exponents is key to simplifying and solving algebraic equations efficiently.

Variable Isolation

Variable isolation is the process of solving an equation by rewriting it to make the unknown variable the subject of the formula. This often involves using inverse operations to "undo" arithmetic operations.

• After simplifying the expression to \(100y = -200\), isolate \(y\) by performing the inverse operation.
  • This means dividing both sides of the equation by \(100\) to "cancel out" the multiplication by \(100\).
• The isolation step results in finding that \(y = \frac{-200}{100}\).
• Finally, simplifying \(\frac{-200}{100}\) gives us the solution \(y = -2\).

Isolating the variable is the ultimate goal when solving equations, as it provides the final answer. Practice with variable isolation enhances problem-solving skills and leads to a clearer understanding of algebraic manipulation.