Problem 4 - Intermediate Algebra

Question

Fill in the blanks. To \(\quad\) a formula for a specified variable means to isolate that variable on one side of the equation, with all other variables and constants on the opposite side.

Step-by-Step Solution

Answer

To **solve** a formula means to isolate a specified variable.

Step 1: Identify the Task

The task is to isolate a specific variable in a formula. Identifying the task helps us understand the purpose of manipulating the equation.

Step 2: Understand the Objective

Isolating a variable means we want that variable to stand alone on one side of the equation, usually on the left, while everything else is moved to the opposite side.

Step 3: Determine the Key Term

The key term that describes the process of isolating a variable in an equation is "solve for" or simply "solve." This is the operation being described by the task.

Step 4: Complete the Sentence

Using the key term identified, we fill in the blank: To **solve** a formula for a specified variable means to isolate that variable on one side of the equation, with all other variables and constants on the opposite side.

Key Concepts

Isolating Variables

When you isolate a variable in an equation, you're essentially trying to "get it by itself" on one side, often the left. This means all the other variables and numbers are moved over to the other side. It's a lot like deciding to have just one thing on one side of a see-saw.

To isolate a variable, you need to reverse the operations that have been applied to it. For example, if a variable is being multiplied by a number, you would divide both sides of the equation by that number to cancel it out. Let's say we have the equation \( 3x = 12 \). To isolate \( x \), we would divide both sides by 3, giving us \( x = 4 \).

Keep in mind:

• Inverse operations are key. Addition becomes subtraction, multiplication becomes division, and so on.
• Whatever you do to one side of the equation, you must do to the other. This keeps the equation balanced, like a see-saw.

Manipulating Equations

Manipulating equations isn't as tricky as it sounds. It simply involves changing the equation's appearance while keeping its essence intact. By essence, we mean the equality between both sides of the equation.

Why manipulate? Sometimes, moving terms around in an equation makes it easier to understand or solve. When manipulating, you should:

• Add or subtract the same number from both sides to move terms.
• Multiply or divide both sides by the same number to change coefficients.

For example, if you start with \( 2x + 5 = 15 \), you can subtract 5 from both sides to get \( 2x = 10 \). Then, divide both sides by 2 to find \( x = 5 \).

This shows how manipulation can simplify and lead to a solution. It’s all a matter of applying basic arithmetic properties carefully and accurately.

Equation Solving Steps

Solving equations often follows a logical sequence of steps. It's like following a recipe to bake something delicious. Each step builds on the last until you reach the goal! The steps generally go like this:

1. **Identify the task.** Understand what variable you need to isolate or find.
2. **Understand the objective.** Know why you are solving and what you hope to achieve, usually having the variable on one side of the equation.
3. **Manipulate the equation.** Use inverse operations to rearrange terms and simplify the equation.
4. **Check your work.** Always plug your solution back into the original equation to make sure it satisfies the condition.

Think of these steps as a reliable guide. By practicing them, you will get better at solving not only small equations but also more complex ones! Remember, like any skill, consistency and practice make perfect.