Solving System Of Equations For P And Q

In mathematics, solving a system of equations is a fundamental skill. This article will guide you through the process of solving a system of two linear equations with two variables, specifically p and q. We'll explore the methods used to find the values of p and q that satisfy both equations simultaneously. We will focus on the substitution method and explain every step in detail so you can easily use it for similar problems.

The System of Equations

We are given the following system of equations:

1. ${ 3p = 2q }$
2. ${ 4p + q + 11 = 0 }$

Our goal is to find the values of p and q that make both of these equations true.

Method 1: Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with a single variable, which can then be easily solved.

Step 1: Solve Equation 1 for q

First, we solve the first equation for q:

${ 3p = 2q }$

Divide both sides by 2:

${ q = \frac{3}{2}p }$

Now we have an expression for q in terms of p.

Step 2: Substitute into Equation 2

Next, we substitute this expression for q into the second equation:

${ 4p + q + 11 = 0 }$

Replace q with ${ \frac{3}{2}p }$:

${ 4p + \frac{3}{2}p + 11 = 0 }$

Step 3: Solve for p

Now we have an equation with only p. To solve for p, we first combine the terms with p:

${ 4p + \frac{3}{2}p = \frac{8}{2}p + \frac{3}{2}p = \frac{11}{2}p }$

So the equation becomes:

${ \frac{11}{2}p + 11 = 0 }$

Subtract 11 from both sides:

${ \frac{11}{2}p = -11 }$

Multiply both sides by ${ \frac{2}{11} }$ to isolate p:

${ p = -11 \times \frac{2}{11} }$

${ p = -2 }$

Thus, we have found that p = -2.

Step 4: Substitute p back into the expression for q

Now that we have the value of p, we can substitute it back into the expression we found for q:

${ q = \frac{3}{2}p }$

Substitute p = -2:

${ q = \frac{3}{2}(-2) }$

${ q = -3 }$

Therefore, we find that q = -3.

Step 5: Verify the Solution

To ensure our solution is correct, we substitute the values of p and q back into the original equations:

Equation 1: ${ 3p = 2q }$

${ 3(-2) = 2(-3) }$

${ -6 = -6 }$ (True)

Equation 2: ${ 4p + q + 11 = 0 }$

${ 4(-2) + (-3) + 11 = 0 }$

${ -8 - 3 + 11 = 0 }$

${ 0 = 0 }$ (True)

Since both equations are true with p = -2 and q = -3, our solution is correct.

In-Depth Explanation of the Substitution Method

Let's delve deeper into the substitution method to understand its mechanics and advantages. The core idea behind this method is to reduce a system of equations into a single equation with one variable. This is achieved by expressing one variable in terms of the other and then replacing that variable in the other equation. The beauty of this method lies in its simplicity and effectiveness, particularly for systems where one equation can be easily solved for one variable.

Choosing the Right Equation and Variable

The first step in the substitution method involves selecting an equation and a variable to solve for. Ideally, you should choose the equation where it is easiest to isolate a variable. This often means looking for a variable with a coefficient of 1 or -1. In our case, the first equation ${ 3p = 2q }$ is straightforward to solve for either p or q. We chose to solve for q in this instance, but solving for p first would also lead to the correct answer. The second equation, ${ 4p + q + 11 = 0 }$, is also relatively simple to solve for q since it has a coefficient of 1.

The Substitution Process

Once you've solved one equation for a variable, the next step is to substitute the resulting expression into the other equation. This is where the magic happens. By substituting, you eliminate one variable from the second equation, leaving you with an equation in just one variable. It’s crucial to substitute the expression correctly, ensuring you replace the variable entirely. A common mistake is only substituting partially or making errors in the algebraic manipulation.

Solving the Single-Variable Equation

After the substitution, you're left with an equation in a single variable. This equation can usually be solved using standard algebraic techniques, such as combining like terms, adding or subtracting constants from both sides, and multiplying or dividing both sides by a coefficient. The goal is to isolate the variable on one side of the equation. In our example, after substituting ${ q = \frac{3}{2}p }$ into the second equation, we obtained ${ 4p + \frac{3}{2}p + 11 = 0 }$, which we then simplified and solved for p.

Back-Substitution

Once you've found the value of one variable, you need to find the value of the other variable. This is done by substituting the value you found back into one of the original equations or, more conveniently, into the expression you derived in the first step. This process is called back-substitution. It’s generally easier to substitute into the expression you found earlier because it's already solved for the other variable. In our example, we substituted p = -2 back into ${ q = \frac{3}{2}p }$ to find q = -3.

Verification: The Crucial Final Step

The final step, and arguably one of the most important, is to verify your solution. This involves substituting the values you found for both variables back into the original equations to ensure they satisfy both. This step helps catch any errors made during the solving process. If the values do not satisfy both equations, you know there’s an error somewhere in your calculations, and you need to go back and check your work. In our case, we verified that p = -2 and q = -3 satisfy both original equations, giving us confidence in our solution.

Alternative Methods for Solving Systems of Equations

While the substitution method is a powerful tool, it's not the only method available for solving systems of equations. Other common methods include the elimination method and graphical methods. Each method has its strengths and weaknesses, and the best choice depends on the specific system of equations you're dealing with.

Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations so that when they are added together, one of the variables is eliminated. This is typically done by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. For example, if you have the system:

${ Ax + By = C }$

${ Dx + Ey = F }$

you might multiply the first equation by E and the second equation by -B so that the y terms will cancel out when the equations are added.

Graphical Methods

Graphical methods involve plotting the equations on a coordinate plane and finding the point of intersection. The coordinates of the point of intersection represent the solution to the system. This method is particularly useful for visualizing the solution and understanding the nature of the system (e.g., whether it has one solution, infinitely many solutions, or no solution). However, graphical methods may not be as precise as algebraic methods, especially when the solutions are not integers.

Comparison of Methods

The substitution method is often preferred when one equation can be easily solved for one variable. The elimination method is particularly effective when the coefficients of one variable are easily made opposites. Graphical methods are useful for visualization but may not be the most precise.

Common Mistakes to Avoid

Solving systems of equations can be tricky, and it’s easy to make mistakes if you’re not careful. Here are some common mistakes to avoid:

1. Incorrect Substitution: Make sure you substitute the expression for the variable into the correct equation and that you replace the variable entirely.
2. Arithmetic Errors: Pay close attention to your arithmetic, especially when dealing with fractions and negative numbers.
3. Forgetting to Distribute: When multiplying an equation by a constant, make sure you distribute the constant to every term in the equation.
4. Not Verifying the Solution: Always verify your solution by substituting the values back into the original equations.
5. Misinterpreting the Question: Ensure you understand what the question is asking. Are you looking for the values of p and q, or is there another requirement?

Conclusion

In this article, we've demonstrated how to solve a system of equations for p and q using the substitution method. We broke down each step, from solving one equation for a variable to substituting and verifying the solution. Understanding these steps and practicing regularly will help you master this fundamental mathematical skill. Remember, the key to success is careful attention to detail and consistent practice. By understanding the underlying principles and avoiding common mistakes, you can confidently tackle any system of equations that comes your way. Whether you choose the substitution method, the elimination method, or a graphical approach, the ability to solve systems of equations is a valuable asset in mathematics and beyond. Keep practicing, and you'll become proficient in no time!