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Algebraic manipulation involves changing the form of an equation to make it easier to solve. It's like rearranging the pieces of a puzzle until they fit. In solving equations, you might multiply, divide, add, or subtract terms to achieve the desired result. These actions help us simplify complex equations and make it possible to solve for a specific variable. For example, in the given equation: \[ p = \frac{r - q}{2} \]we used multiplication to eliminate the fraction. By multiplying both sides by 2, we made the equation simpler: \[ 2p = r - q \]Learning to identify the right operation to apply is key to mastering algebraic manipulations. This skill will help you gain confidence in solving various types of mathematical problems.

Isolating variables means we want to get a specific variable by itself on one side of the equation. This often involves a series of algebraic manipulations to move all terms involving the variable to one side and all other terms to the other side. In the exercise, after multiplying by 2, we wanted to solve for q. The original equation became: \[ 2p = r - q \]To isolate q, we first moved -q to the other side by adding q to both sides, resulting in: \[ 2p + q = r \]Then, we further isolated q by subtracting 2p from both sides: \[ q = r - 2p \]When isolating variables, it's essential to perform the same operation on both sides of the equation to maintain equality. This ensures that the change in the equation still reflects the original relationship between the variables.

Solving equations involves a clear and systematic approach. Here's a breakdown of the steps taken in our example problem:

• Step 1: Multiply both sides by 2 to eliminate the fraction: \[ 2p = r - q \]
• Step 2: Add q to both sides to move q to one side: \[ 2p + q = r \]
• Step 3: Subtract 2p from both sides to isolate q: \[ q = r - 2p \]

Following these ordered steps ensures you apply logical and mathematical principles correctly each time. Each step should bring you closer to isolating the variable and solving the equation. This systematic approach can be applied to various problems to help you solve equations efficiently and accurately.