91Ó°ÊÓ

Isolating variables is a fundamental concept in algebra. It involves manipulating an equation to get the variable of interest by itself, typically on one side of the equation. Let’s take a closer look at how we isolated the variable in the given problem.

We started with the equation: \[ x = \frac{3}{1-y} \].

To isolate the variable \( y \), we executed several key steps:

• Step 1: Multiply both sides by the denominator to eliminate the fraction. This gave us \( x(1-y) = 3 \).
• Step 2: Distribute \( x \), resulting in \( x - xy = 3 \).
• Step 3: Rearrange the equation to isolate the term containing \( y \), leading to \( -xy = 3 - x \).
• Step 4: Finally, divide by \( -x \) to solve for \( y \), simplifying to \( y = \frac{x - 3}{x} \).

Each action taken ensures we keep the equality balanced, emphasizing how crucial it is to perform the same operations on both sides. Proper isolation of variables is key to solving more complex problems in algebra.

The distributive property is an essential algebraic principle. It allows us to multiply a single term across terms inside parentheses. This property makes simplifying expressions easier and solving equations more straightforward.

In the equation \( x = \frac{3}{1-y} \), after multiplying both sides by \( 1-y \), we used the distributive property to expand \( x(1-y) \).

• This gave us \( x - xy \) by distributing \( x \) to both 1 and \( -y \).

The distributive property can be described by the formula: \[ a(b+c) = ab + ac \]. It ensures precision in expansion, and without it, more complex algebraic manipulations would be daunting.

Motion problems in algebra typically deal with scenarios involving distance, speed, and time. Though the original exercise was not a typical motion problem, the same algebraic skills are essential.

In motion problems, we often use formulas like:

\( \text{Distance} = \text{Speed} \times \text{Time} \).

Solving a motion problem means isolating one of these variables depending on the given information. For example, if you know the distance and speed, you solve for time:

\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

• First, identify the knowns and unknowns.
• Next, write down the algebraic equation relating them.
• Isolate the variable you need to find using isolation and distribution techniques as required.

Mastering these algebraic strategies in simpler contexts like isolating variables and using the distributive property will build a strong foundation for solving motion problems effectively.