Problem 36 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ 6 w-4 x=3 w+5 x, \text { for } w $$

Step-by-Step Solution

Answer

Answer: w = 3x

Step 1: Isolate w terms on one side of the equation

To isolate w terms on one side of the equation, we will first move the 3w term from the right side to the left side by subtracting 3w from both sides. This will give us: $$ 6w - 4x - 3w = 5x $$ Then, simplify the left side by combining like terms: $$ (6w - 3w) - 4x = 3w - 4x = 5x $$

Step 2: Isolate x terms on the other side of the equation

Now we will move the -4x term from the left side to the right side by adding 4x to both sides: $$ 3w = 5x + 4x $$ Then, simplify the right side by combining like terms: $$ 3w = 9x $$

Step 3: Solve for w

Finally, to solve for w, we will divide both sides of the equation by 3: $$ \frac{3w}{3} = \frac{9x}{3} $$ This simplifies to: $$ w = 3x $$ So the solution is: $$ \boxed{w=3x} $$

Key Concepts

Isolation of Variables

When solving linear equations, one of the primary goals is the isolation of variables. This means getting the variable you're solving for alone on one side of the equation. To achieve this, you often need to rearrange the terms. In our example, we aim to isolate the variable \( w \). We do this by carefully moving terms that contain \( w \) to one side of the equation. This starts by subtracting \( 3w \) from both sides, which allows us to see clearly which terms are associated with \( w \) and which are not.

• Subtracting or adding terms: Move terms that contain the variable to be isolated.
• Choose one side for the variable: Gather all the terms involving the target variable on one side.

By following these steps, you efficiently position the equation to be solved, ensuring all components with the desired variable are grouped together.

Algebraic Manipulation

Algebraic manipulation involves rearranging the equation through operations such as addition, subtraction, multiplication, etc., to simplify it. After isolating \( w \) on one side, algebraic manipulation helps simplify the equation further and solve for the variable. This task may include moving other terms or performing operations to simplify both sides of the equation.

• Perform opposite operations: To move terms, do the opposite operation; subtract if added, add if subtracted.
• Simplify step-by-step: Always simplify as much as possible once terms are rearranged.

In the given exercise, we started by subtracting \( 3w \) and subsequently added \( 4x \) to both sides. Such skills are vital in algebra as they simplify complex problems into more manageable forms.

Combining Like Terms

Combining like terms is an essential algebraic process that simplifies equations and expressions. Like terms are terms that have the same variable raised to the same power. In the context of our problem, once the \( w \) terms are on one side, and the \( x \) terms are on the other, you combine them to make the equation simpler.

• Identify like terms: Terms that share the same variable and exponent.
• Add or subtract coefficients: Combine the numeric part of the terms.

In our equation, \( 6w - 3w \) simplifies to \( 3w \), while \( 5x + 4x \) becomes \( 9x \). Combining these terms reduces the clutter in an equation and makes it easier to isolate variables, thus reaching a solution more efficiently.