Substituting into the First Equation

Oct 13, 2025 by windailys.com 37 views

Introduction

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore a step-by-step approach to solving systems of equations using two different methods: elimination and substitution. We will use a specific system of equations as an example to illustrate these methods.

The System of Equations

Tyler is working with the following system of equations:

\begin{cases} 4p + 2q = 62 \\ 8p - q = 59 \end{cases}

Method 1: Elimination

One way to solve this system of equations is to eliminate one of the variables by multiplying one of the equations by a suitable constant and then subtracting the other equation. Tyler can think of two ways to eliminate a variable and solve the system:

Option 1: Multiply the First Equation by 2

If Tyler multiplies the first equation by 2, he gets:

8p + 4q = 124

Now, he can subtract the second equation from this new equation to eliminate the variable $p$ :

\begin{align*} (8p + 4q) - (8p - q) &= 124 - 59 \\ 5q &= 65 \\ q &= 13 \end{align*}

Option 2: Multiply the Second Equation by 2

Alternatively, Tyler can multiply the second equation by 2 to get:

16p - 2q = 118

Now, he can add the first equation to this new equation to eliminate the variable $p$ :

\begin{align*} (4p + 2q) + (16p - 2q) &= 62 + 118 \\ 20p &= 180 \\ p &= 9 \end{align*}

Method 2: Substitution

Another way to solve this system of equations is to use the substitution method. Tyler can solve one of the equations for one of the variables and then substitute that expression into the other equation.

Solving for $p$

Let's solve the second equation for $p$ :

8p - q = 59

8p = 59 + q

p = \frac{59 + q}{8}

Substituting into the First Equation

Now, Tyler can substitute this expression for $p$ into the first equation:

4\left(\frac{59 + q}{8}\right) + 2q = 62

\begin{align*} \frac{236 + 4q}{8} + 2q &= 62 \\ 236 + 4q + 16q &= 496 \\ 20q &= 260 \\ q &= 13 \end{align*}

Finding the Value of $p$

Now that Tyler has found the value of $q$ , he can substitute this value into one of the original equations to find the value of $p$ . Let's use the second equation:

8p - q = 59

8p - 13 = 59

8p = 72

p = 9

Conclusion

In this article, we have explored two different methods for solving systems of equations: elimination and substitution. We have used a specific system of equations as an example to illustrate these methods. By following these steps, Tyler was able to solve the system of equations and find the values of the variables $p$ and $q$ . These methods can be applied to a wide range of systems of equations, and are an essential tool for any mathematician or scientist.

Additional Resources

For more information on solving systems of equations, check out the following resources:

Khan Academy: Solving Systems of Equations
Mathway: Solving Systems of Equations
Wolfram Alpha: Solving Systems of Equations

Practice Problems

Try solving the following system of equations using the elimination and substitution methods:

\begin{cases} 2x + 3y = 12 \\ x - 2y = -3 \end{cases}

Answer Key

The solution to the system of equations is:

x = 5, y = 1

Final Thoughts

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. In other words, it is a collection of equations that are related to each other through the variables.

Q: Why do we need to solve systems of equations?

A: Solving systems of equations is an essential tool in mathematics and science. It allows us to find the values of variables that satisfy multiple equations simultaneously, which is crucial in many real-world applications, such as physics, engineering, economics, and computer science.

Q: What are the two main methods for solving systems of equations?

A: The two main methods for solving systems of equations are:

Elimination Method: This method involves eliminating one of the variables by adding or subtracting the equations.
Substitution Method: This method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.

Q: How do I choose which method to use?

A: The choice of method depends on the specific system of equations and the variables involved. If the coefficients of the variables are easy to work with, the elimination method may be more straightforward. If the coefficients are more complicated, the substitution method may be more suitable.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

Not checking the solution: Make sure to check the solution by plugging it back into the original equations.
Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
Not using the correct method: Make sure to use the correct method for the specific system of equations.

Q: How do I know if a system of equations has a unique solution?

A: A system of equations has a unique solution if it has a single solution that satisfies both equations. If the system has multiple solutions or no solutions, it is said to be inconsistent.

Q: Can a system of equations have multiple solutions?

A: Yes, a system of equations can have multiple solutions. This occurs when the equations are dependent, meaning that one equation is a multiple of the other.

Q: Can a system of equations have no solutions?

A: Yes, a system of equations can have no solutions. This occurs when the equations are inconsistent, meaning that they contradict each other.

Q: How do I graph a system of equations?

A: To graph a system of equations, plot the equations on a coordinate plane and find the point of intersection. The point of intersection represents the solution to the system.

Q: What are some real-world applications of solving systems of equations?

A: Solving systems of equations has many real-world applications, including:

Physics: Solving systems of equations is used to describe the motion of objects in physics.
Engineering: Solving systems of equations is used to design and optimize systems in engineering.
Economics: Solving systems of equations is used to model economic systems and make predictions.
Computer Science: Solving systems of equations is used in computer science to solve problems in machine learning and optimization.

Conclusion

Solving systems of equations is a fundamental concept in mathematics that has many real-world applications. By understanding the two main methods for solving systems of equations (elimination and substitution) and avoiding common mistakes, you can become proficient in solving systems of equations and apply these skills to a wide range of problems.