Problem 90 - College Algebra with Modeling and Visualization

Question

Solve the equation for the specified variable. $$ y=4-(8-2 x) \text { for } x $$

Step-by-Step Solution

Answer

x = \frac{y + 4}{2}.

Step 1: Simplify the Right Side

First, simplify the expression on the right side of the equation. Start by handling the parentheses and subtraction: \( y = 4 - (8 - 2x) = 4 - 8 + 2x \). This simplifies to \( y = -4 + 2x \).

Step 2: Isolate the Term with the Variable

Move all the terms involving \( x \) to one side and constants to the other side. Rearrange the equation: \( y + 4 = 2x \).

Step 3: Solve for the Variable

Isolate \( x \) by dividing both sides of the equation by 2: \( x = \frac{y + 4}{2} \).

Key Concepts

Variable Isolation

In algebra, isolating the variable is a crucial step in solving equations. The main goal is to have the variable on one side of the equation and everything else on the other. This involves manipulating the equation through addition, subtraction, multiplication, or division.

When isolating the variable, consider all of the terms that include the variable you're solving for, like in our original exercise solving for \(x\). To move terms from one side of the equation to the other, you apply the inverse operation. For instance, if a term is subtracted on one side, you add it to both sides of the equation.

• In the original problem, the equation became \(y + 4 = 2x\) by moving the constant terms.
• The goal was to leave \(x\) alone on one side. This was done by dividing both sides by 2.

This process ensures that you systematically work towards having \(x\) isolated, which makes finding its value straightforward.

Equation Simplification

Simplifying an equation helps you see it more clearly and solve it more effectively. Simplification involves reducing complexity in an equation by combining like terms and dealing with parentheses and operations.

In our provided exercise, simplification involved clearing parentheses and then simplifying the terms:

• First, handle the inner subtraction: \(8 - 2x\). This keeps the equation balanced.
• Next, remove the parentheses by distributing any subtraction or addition across the terms inside.
• In this case, \(y = 4 - (8 - 2x)\) becomes \(y = 4 - 8 + 2x\), further simplified to \(y = -4 + 2x\).

The goal of simplification is to make the equation easier to work with so you can quickly and efficiently isolate the variable.

Problem Solving Steps

Problem-solving in algebra often follows a structured approach. This method provides clarity and makes tackling complex problems manageable. Here are the typical steps used in solving equations like our exercise:

• **Identify Terms Involving Variables:** Determine which terms contain the variable you need to solve for.
• **Simplify If Necessary:** Before isolating the variable, reduce any complex parts of the equation. This can include distributing or combining like terms.
• **Rearrange:** If needed, rearrange the equation so that all terms containing the variable are on one side.
• **Isolate the Variable:** Use inverse operations to isolate the variable. For example, in the exercise, dividing both sides by 2 was necessary.

Following these steps methodically can reduce errors and improve your problem-solving skills in algebra.