Solving For P And Q In The Equations 2p + Q = 33 And P = 5q
In mathematics, solving a system of equations is a fundamental skill. It allows us to determine the values of unknown variables that satisfy a set of given equations. This article delves into the process of solving a system of two equations with two variables, specifically focusing on the equations 2p + q = 33 and p = 5q. We will explore the techniques involved in finding the values of p and q that simultaneously satisfy both equations. Understanding these methods is crucial for various applications in mathematics, science, and engineering. Let's embark on this journey of mathematical problem-solving.
Understanding the Problem: System of Equations
Before diving into the solution, let's clarify what we mean by a system of equations. A system of equations is a collection of two or more equations that involve the same set of variables. The goal is to find the values of these variables that make all the equations true simultaneously. In our case, we have two equations:
- 2p + q = 33
- p = 5q
These equations involve two variables, p and q. Our objective is to find the specific values of p and q that satisfy both equations. This means that when we substitute these values into both equations, the equations hold true. Solving systems of equations is a crucial skill in algebra and has numerous applications in various fields, including physics, economics, and computer science.
Method 1: Substitution Method
The substitution method is a powerful technique for solving systems of equations. The core idea behind this method is to express one variable in terms of the other from one equation and then substitute this expression into the other equation. This process reduces the system to a single equation with one variable, which can then be easily solved. Let's apply this method to our system:
- 2p + q = 33
- p = 5q
Notice that the second equation, p = 5q, already expresses p in terms of q. This makes it convenient to substitute this expression into the first equation. Replacing p in the first equation with 5q, we get:
2(5q) + q = 33
Now we have a single equation with only one variable, q. Simplifying this equation:
10q + q = 33
11q = 33
Dividing both sides by 11, we find:
q = 3
Now that we have found the value of q, we can substitute it back into either of the original equations to find the value of p. It's often easier to use the simpler equation, which in this case is p = 5q. Substituting q = 3, we get:
p = 5(3)
p = 15
Therefore, the solution to the system of equations is p = 15 and q = 3. This method is particularly effective when one of the equations is already solved for one variable in terms of the other.
Method 2: Elimination Method
The elimination method, also known as the addition or subtraction method, is another effective technique for solving systems of equations. The key idea behind this method is to manipulate the equations in such a way that when they are added or subtracted, one of the variables is eliminated. This leaves us with a single equation with one variable, which can be solved directly. Let's apply this method to our system:
- 2p + q = 33
- p = 5q
To use the elimination method effectively, we need to rewrite the second equation so that the variables are aligned as in the first equation. We can subtract 5q from both sides of the second equation to get:
p - 5q = 0
Now our system of equations looks like this:
- 2p + q = 33
- p - 5q = 0
To eliminate p, we can multiply the second equation by -2:
-2(p - 5q) = -2(0)
-2p + 10q = 0
Now we have the following system:
- 2p + q = 33
- -2p + 10q = 0
Adding the two equations together, we eliminate p:
(2p + q) + (-2p + 10q) = 33 + 0
11q = 33
Dividing both sides by 11, we get:
q = 3
Now that we have the value of q, we can substitute it back into either of the original equations to find p. Using the second equation, p = 5q, we substitute q = 3:
p = 5(3)
p = 15
Thus, the solution to the system of equations, using the elimination method, is p = 15 and q = 3. The elimination method is particularly useful when the coefficients of one of the variables are the same or can be easily made the same by multiplication.
Verification of the Solution
After finding the values of p and q, it's crucial to verify that these values indeed satisfy both equations in the system. This step ensures that our solution is correct and that we haven't made any errors in our calculations. Let's verify our solution, p = 15 and q = 3, for the given system of equations:
- 2p + q = 33
- p = 5q
Verification for Equation 1:
Substitute p = 15 and q = 3 into the first equation:
2(15) + 3 = 33
30 + 3 = 33
33 = 33
The first equation holds true.
Verification for Equation 2:
Substitute p = 15 and q = 3 into the second equation:
15 = 5(3)
15 = 15
The second equation also holds true.
Since both equations are satisfied by p = 15 and q = 3, we can confidently conclude that this is the correct solution to the system of equations. Verification is an essential step in problem-solving, as it helps to catch any potential errors and ensures the accuracy of the solution.
Real-World Applications of Solving Systems of Equations
Solving systems of equations is not just a theoretical exercise; it has numerous practical applications in various real-world scenarios. These applications span across diverse fields, including science, engineering, economics, and computer science. Understanding how to solve these systems allows us to model and analyze complex situations, making informed decisions and predictions. Let's explore some examples of real-world applications:
1. Mixture Problems
Mixture problems often involve combining two or more substances with different concentrations or properties to obtain a desired mixture. These problems can be modeled using systems of equations. For instance, consider a scenario where you need to mix two solutions with different concentrations of a chemical to obtain a solution with a specific concentration. By setting up equations representing the amounts of each solution and the total amount of the chemical, you can solve for the required quantities of each solution.
2. Investment and Finance
Financial calculations frequently involve systems of equations. For example, when dealing with investments, you might need to determine the amount to invest in different accounts with varying interest rates to achieve a specific return. Similarly, loan calculations, break-even analysis, and portfolio management often require solving systems of equations to determine optimal strategies and outcomes. Solving these equations helps in making informed financial decisions.
3. Supply and Demand
In economics, the equilibrium price and quantity of a product in a market are determined by the intersection of the supply and demand curves. These curves can be represented by equations, and the point of intersection is the solution to the system of equations. Solving this system allows economists to predict market behavior, understand the impact of various factors on prices and quantities, and develop effective economic policies.
4. Network Analysis
Network analysis, such as in electrical circuits or transportation networks, often involves solving systems of equations. In electrical circuits, Kirchhoff's laws provide a set of equations that relate the currents and voltages in the circuit. Solving these equations allows engineers to determine the currents flowing through different parts of the circuit and the voltage drops across various components. Similarly, in transportation networks, systems of equations can be used to model traffic flow and optimize routes.
5. Physics and Engineering
Many problems in physics and engineering involve multiple variables and constraints, which can be modeled using systems of equations. For example, in mechanics, the forces acting on an object can be analyzed using systems of equations to determine the object's motion. In structural engineering, systems of equations are used to analyze the stresses and strains in structures. These applications highlight the importance of solving systems of equations in scientific and engineering problem-solving.
Conclusion
In this article, we have explored the process of solving a system of two equations with two variables: 2p + q = 33 and p = 5q. We discussed two common methods, the substitution method and the elimination method, both of which led us to the solution p = 15 and q = 3. We also emphasized the importance of verifying the solution to ensure accuracy. Furthermore, we highlighted the diverse real-world applications of solving systems of equations, demonstrating its relevance in various fields. Mastering the techniques for solving systems of equations is an invaluable skill that empowers us to tackle complex problems and make informed decisions in numerous contexts.