Question

Given \(p / q=r\), solve for \(q\).

Short answer

The short answer for solving \(q\) in the given equation \(p / q = r\) is: \(q = \frac{p}{r}\).

Step-by-step solution

Write down the given equation

We have the equation \(p / q = r\).

Multiply both sides by \(q\) to isolate the term with \(q\)

To get rid of the division by \(q\), we can multiply both sides of the equation by \(q\). This will give us the equation: \(q \cdot (p / q) = r \cdot q\).

Simplify the equation

Since we multiplied both sides by \(q\), we can now simplify the equation. On the left side of the equation, \(q\) cancels out: \(\frac{p}{q} \cdot q = p\). So now the equation becomes: \(p = r \cdot q\).

Solve for \(q\) by dividing both sides by \(r\)

To get \(q\) by itself, we need to remove \(r\) from the right side of the equation. We do this by dividing both sides of the equation by \(r\): \(\frac{p}{r} = q\).

Write the final answer

So, the final solution for \(q\) is: \(q = \frac{p}{r}\).

Key concepts

Algebraic Manipulation

Understanding algebraic manipulation is key to solving equations efficiently. When faced with an equation like the given example, \( p / q = r \), the objective is to modify or 'manipulate' the equation using algebraic operations to isolate and solve for the desired variable—in this case, \( q \). Algebraic manipulation includes performing the same operation to both sides of the equation, ensuring that it remains balanced. Operations such as addition, subtraction, multiplication, and division are tools you can use to transform the equation into a more solvable form.

For example, multiplying both sides of \( p / q = r \) by \( q \) can help eliminate the fraction, which is exactly what was done in the second step of the solution. Such techniques are not only used for equations with fractions but are a foundational aspect of solving various algebraic expressions.

Isolation of Variables

The goal of isolating the variable is to rewrite an equation so that one variable, usually the one we are solving for, is on one side of the equation and all other variables and constants are on the other side. In the given exercise, the variable we needed to isolate was \( q \).

Our starting point was \( p / q = r \), and by multiplying both sides by \( q \) and then dividing both sides by \( r \), we moved everything except for \( q \) to the other side of the equation. This process ultimately led us to the answer: \( q = p / r \). Isolating the variable is a fundamental technique because it lays the groundwork for solving the equation, making it clearer what value the variable holds.

Equation Simplification

To simplify an equation means to reduce it to its simplest form, making it easier to solve. Simplification often involves eliminating unnecessary terms, canceling out factors, and combining like terms. In our exercise, simplification occurred after multiplying by \( q \), which canceled out the denominator, leading to \( p = r \cdot q \).

It's essential to recognize that simplifying an equation does not change its solutions; rather, it makes the solutions more apparent. Think of simplification as a way to clear the clutter, allowing you to see the 'bare bones' of the equation. Simplification techniques are not only applied to solving for a variable but also for making complex algebraic expressions more manageable and understandable.