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Equation manipulation is a fundamental skill in algebra that involves changing the form of an equation to isolate a particular variable. This often involves applying various algebraic operations like addition, subtraction, multiplication, or division.

To manipulate equations effectively, it's important to understand the properties of equality which state that you can perform the same operation on both sides of the equation without changing its truth value. For example:

• Adding or subtracting the same number from both sides.
• Multiplying or dividing both sides by the same non-zero number.

In the given problem, we started with the equation:\(p=\frac{h}{2}(q+r)\).

The goal was to solve for \(q\). The first step involved reversing the multiplication by \(\frac{h}{2}\) through multiplying both sides by \(\frac{2}{h}\). This isolated the terms involving \(q\), making it easier to solve for the desired variable.

Solving for a variable means finding the value of that variable in an equation. To do this, we need to isolate the variable on one side of the equation. In the exercise, we wanted to solve for \(q\) from the equation:
\( \frac{2p}{h} = q + r \).

The process involves these steps:

• `Identify` what you want to solve for.
• `Use` inverse operations to remove other terms from around the variable.
• `Simplify` the equation after each step.

After manipulating the equation into the form \( \frac{2p}{h} = q + r \), we needed to isolate \(q\). By subtracting \(r\) from both sides, we remove it from the side of the equation that contains \(q\), achieving:
\( q = \frac{2p}{h} - r \).

This shows \(q\) expressed in terms of \(p\), \(h\), and \(r\).

Algebraic expressions involve numbers, variables, and operations (such as addition, subtraction, multiplication, and division). Understanding algebraic expressions is central to solving equations since they form the building blocks of equations.

In the exercise stated, two key expressions played a role:

• \(\frac{h}{2}(q+r)\) on the right side of the initial equation, which combines multiplication and addition operations.
• \(\frac{2p}{h} - r\), the simplified expression representing \(q\) in terms of other variables.

To navigate these expressions, focus on handling each operation step-by-step, as algebraic expressions are simplified through systematic approaches. Knowing that each operation can be reversed or counterbalanced is crucial for working through more complex problems. Expressions allow you to represent relationships between quantities, leading to a deeper understanding of how changes in one variable may affect others.