Question

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

Short answer

The short answer to the given equation \( \frac{p}{q} = r \) is to multiply both sides by \(q\) to isolate \(p\), resulting in the solution: \( p = rq \).

Step-by-step solution

Write down the given equation

We start with the given equation: \( \frac{p}{q} = r \)

Isolate p

To isolate \(p\), we need to eliminate the fraction. Since \(p\) is divided by \(q\) in the fraction, to eliminate the denominator, we can multiply both sides of the equation by \(q\). This operation gives us: \( p = rq \)

Final answer

After isolating \(p\), the solution to the given equation \(p / q = r\), is: \[ p = rq \]

Key concepts

Equation solving

Solving equations is a fundamental skill in algebra that helps you find the value of an unknown variable. An equation is a statement that asserts the equality of two expressions. In our exercise, the given equation was \( \frac{p}{q} = r \). The main goal is to determine the value of \( p \) by manipulating the equation to isolate \( p \) on one side of the equation. Solving equations often involves performing the same operation on both sides to keep the equation balanced. This step-by-step process ensures that you maintain the equality and find a valid solution for the unknown variable.

Isolation of variables

Isolation of variables is key when solving equations. It involves rearranging an equation so that the variable of interest appears alone on one side. This process is essential for determining the value of the unknown variable. Let's consider the equation \( \frac{p}{q} = r \). To isolate \( p \), multiplication is used to eliminate the fraction.

• Multiply both sides of the equation by \( q \) to remove it from the denominator.
• The resulting equation is \( p = rq \).

This method of isolating \( p \) ensures that you can clearly see the relationship between \( p \), \( q \), and \( r \), allowing for a straightforward computation of \( p \). By carefully isolating variables, you clarify the equation structure and simplify the problem-solving process.

Multiplication property of equality

The multiplication property of equality is a crucial concept used to manipulate equations while maintaining equality. It states that if you multiply both sides of an equation by the same non-zero number, the two sides remain equal. In our exercise, this property was instrumental in isolating \( p \) from the equation \( \frac{p}{q} = r \). After noticing the fraction, we used the multiplication property by:

• Multiplying both sides by \( q \).
• This operation transforms the equation, simplifying \( \frac{p}{q} \times q = rq \), which results in \( p = rq \).

Employing the multiplication property allows us to eliminate fractions or coefficients hindering the isolation of the unknown variable. This operation not only simplifies complex relationships but also reinforces the balance needed in equations, ensuring valid solutions.