Solving The Relationship Between Variables P And Q A Step-by-Step Guide
This article delves into a fascinating mathematical problem that explores the relationship between two variables, p and q. The problem states that p is partly constant and partly varies as q. This means that p can be expressed as the sum of two components: a constant term and a term that is proportional to q. We are given two sets of values for p and q: when q = 5, p = 7, and when q = 7, p = 45. Our objective is twofold: first, to determine the precise relationship between p and q, and second, to find the value of p when q is 15. This problem is a classic example of direct and partial variation, which is a fundamental concept in algebra and has applications in various fields, including physics, engineering, and economics. Understanding how to solve this type of problem is crucial for developing strong mathematical reasoning and problem-solving skills. In this detailed exploration, we will break down the problem step by step, providing a clear and comprehensive explanation of the solution process. We will start by establishing the general equation that represents the relationship between p and q, and then we will use the given data to solve for the unknown constants. Finally, we will use the derived relationship to calculate the value of p when q is 15. Through this process, we aim to not only provide a solution to the problem but also to enhance your understanding of direct and partial variation and its applications.
(a) Unveiling the Relationship Between p and q
The first part of the problem requires us to find the relationship between the variables p and q. As stated earlier, p is partly constant and partly varies as q. This can be mathematically expressed as:
p = k + mq
where k is the constant term and m is the constant of proportionality. Our goal is to determine the values of k and m. To do this, we will use the given information: when q = 5, p = 7, and when q = 7, p = 45. Substituting these values into the equation, we get two equations:
7 = k + 5m (Equation 1)
45 = k + 7m (Equation 2)
We now have a system of two linear equations with two unknowns. There are several methods to solve such systems, including substitution, elimination, and matrix methods. In this case, the elimination method is particularly straightforward. We can subtract Equation 1 from Equation 2 to eliminate k:
45 - 7 = (k + 7m) - (k + 5m)
38 = 2m
Dividing both sides by 2, we find:
m = 19
Now that we have the value of m, we can substitute it back into either Equation 1 or Equation 2 to solve for k. Let's use Equation 1:
7 = k + 5(19)
7 = k + 95
Subtracting 95 from both sides, we get:
k = -88
Therefore, we have found the values of k and m: k = -88 and m = 19. Substituting these values back into the general equation, we obtain the specific relationship between p and q:
p = -88 + 19q**
This equation represents the precise relationship between p and q for this particular problem. It tells us that p is equal to -88 plus 19 times q. This linear relationship confirms the problem's initial statement that p is partly constant (-88) and partly varies as q (19q). Understanding how to derive this relationship is crucial for solving similar problems involving direct and partial variation. The ability to translate a verbal description of a relationship into a mathematical equation is a fundamental skill in algebra and mathematical modeling.
(b) Finding p When q = 15
Now that we have established the relationship between p and q as p = -88 + 19q, the second part of the problem asks us to find the value of p when q = 15. This is a straightforward application of the equation we derived. To find p, we simply substitute q = 15 into the equation:
p = -88 + 19(15)
First, we calculate the product of 19 and 15:
19 * 15 = 285
Now, we substitute this value back into the equation:
p = -88 + 285
Finally, we perform the addition:
p = 197
Therefore, when q = 15, the value of p is 197. This result demonstrates how we can use the derived relationship between variables to predict the value of one variable given the value of the other. This is a powerful application of mathematical modeling and is used extensively in various fields to make predictions and understand the behavior of systems. The process of substituting a value into an equation and solving for the unknown variable is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. In this case, we successfully used the relationship between p and q to determine the value of p when q is 15, showcasing the practical application of direct and partial variation.
In conclusion, we have successfully solved the problem by first determining the relationship between p and q as p = -88 + 19q and then using this relationship to find the value of p when q = 15, which is 197. This problem highlights the importance of understanding direct and partial variation and how to translate verbal descriptions into mathematical equations. The ability to solve such problems is a valuable skill in mathematics and has applications in various real-world scenarios. This detailed explanation provides a clear understanding of the solution process and reinforces the concepts of direct and partial variation.
