The value of \(m\) is directly proportional to the value of \(p .\) When \(m=2, p=6 .\) What is \(m\) when \(p=9 ?\) F. \(\frac{1}{3}\) G. \(\frac{4}{3}\) H. 3 J. 5 K. 27

Understanding the constant of proportionality is crucial when exploring direct relationships between variables. In direct proportionality, two variables change at the same rate. This constant rate is what we refer to as the constant of proportionality.

For example, in the relationship between the values of \(m\) and \(p\), where \(m\) is directly proportional to \(p\), this constant tells us how much \(m\) changes in response to a change in \(p\). It is represented by the symbol \(k\), and the equation modeling their relationship is \(m = k \cdot p\).

Consider a scenario where you have been given certain values of \(m\) and \(p\), like \(m=2\) and \(p=6\). You can insert these into the equation to solve for the constant of proportionality, which is, in this case, \(k = \frac{1}{3}\). The value of \(k\) remains fixed and can be used to predict the value of \(m\) for any other given value of \(p\), demonstrating the power of understanding this constant in various contexts such as physics, economics, and even daily life tasks.

Algebraic equations are the foundation of a vast range of mathematical problem-solving. An algebraic equation is a statement indicating the equality of two algebraic expressions. These can include one or multiple variables and constants.

The equation \(m = k \cdot p\) from our proportionality problem is an algebraic equation involving the constant \(k\) and variables \(m\) and \(p\). In this instance, the equation expresses a direct relationship between \(m\) and \(p\), which helps us understand how to manipulate and utilize these relationships to solve for unknowns.

To use such an equation effectively, we input known quantities to derive the unknown ones. Through methods such as substitution, you can uncover the value of any variable provided you have sufficient information. For students, mastering algebraic equations is imperative, as they are the building blocks for more complex mathematical concepts and are indispensable in various fields of study.

Solving for variables is a fundamental skill in algebra, which involves finding the values of unknown quantities represented by symbols. When dealing with direct proportions, solving for a variable requires isolating it on one side of the equation.

In the problem provided, once the constant of proportionality \(k\) is found, we use it to determine the value of \(m\) when \(p=9\). You solve for \(m\) by substituting the known values into the proportionality equation: \(m = \frac{1}{3} \cdot 9\). After simplifying, \(m\) equals 3, which is the sought value.

Such problems demand careful attention to the steps: identifying the correct equation to use, substituting the proper values for variables, and then performing the necessary arithmetic operations. Through practice and understanding the underlying concepts, students become adept at solving for variables, which is instrumental not only in algebra but in the application of mathematics to real-world problems.