CFA 3.4 / Q6Directions: Solve The Linear System Using Substitution. Enter Your Answer As An Ordered Pair, $ (x, Y) . . . \begin{cases} y = X + 2 \ 2x + Y = 8 \end{cases} $Solution: $ \square $

Introduction

Linear systems are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving linear systems using substitution, a method that involves expressing one variable in terms of the other and then substituting it into the other equation. We will use the given problem as a case study to illustrate this method.

The Problem

The problem we will be solving is a linear system consisting of two equations:

$$\begin{cases} y = x + 2 \\ 2x + y = 8 \end{cases}$$

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

Step 1: Express One Variable in Terms of the Other

The first equation is already solved for $y$, so we can express $y$ in terms of $x$:

$$y = x + 2$$

This equation tells us that $y$ is equal to $x$ plus 2.

Step 2: Substitute the Expression into the Other Equation

Now that we have expressed $y$ in terms of $x$, we can substitute this expression into the second equation:

$$2x + (x + 2) = 8$$

Step 3: Simplify the Equation

We can simplify the equation by combining like terms:

$$3x + 2 = 8$$

Step 4: Solve for $x$

Now we can solve for $x$ by subtracting 2 from both sides of the equation:

$$3x = 6$$

Dividing both sides by 3, we get:

$$x = 2$$

Step 5: Find the Value of $y$

Now that we have found the value of $x$, we can substitute it into the first equation to find the value of $y$:

$$y = x + 2$$

$$y = 2 + 2$$

$$y = 4$$

Conclusion

We have successfully solved the linear system using substitution. The solution is $x = 2$ and $y = 4$. This means that the point $(2, 4)$ satisfies both equations.

Why Substitution Works

Substitution works because it allows us to eliminate one variable and solve for the other. By expressing one variable in terms of the other, we can substitute it into the other equation and solve for the remaining variable. This method is particularly useful when one of the equations is already solved for one of the variables.

Real-World Applications

Solving linear systems using substitution has many real-world applications. For example, in economics, linear systems can be used to model supply and demand curves. In engineering, linear systems can be used to model electrical circuits and mechanical systems. In computer science, linear systems can be used to solve problems in machine learning and data analysis.

Tips and Tricks

Here are some tips and tricks to help you solve linear systems using substitution:

• Make sure to express one variable in terms of the other before substituting it into the other equation.
• Simplify the equation as much as possible before solving for the variable.
• Check your solution by plugging it back into both equations.

Conclusion

Q: What is substitution in linear systems?

A: Substitution is a method used to solve linear systems by expressing one variable in terms of the other and then substituting it into the other equation.

Q: Why is substitution useful?

A: Substitution is useful because it allows us to eliminate one variable and solve for the other. This method is particularly useful when one of the equations is already solved for one of the variables.

Q: How do I know which variable to express in terms of the other?

A: You can choose either variable to express in terms of the other. However, it's often easier to express the variable that appears in only one of the equations.

Q: What if I have a system with three or more equations?

A: In that case, you can use substitution to solve for two variables, and then use the third equation to solve for the remaining variable.

Q: Can I use substitution to solve systems with fractions or decimals?

A: Yes, you can use substitution to solve systems with fractions or decimals. Just make sure to simplify the equation as much as possible before solving for the variable.

Q: What if I get stuck or make a mistake?

A: Don't worry! It's easy to get stuck or make a mistake when solving linear systems. Just take a step back, re-read the problem, and try again. You can also ask for help from a teacher or tutor.

Q: Are there any other methods for solving linear systems?

A: Yes, there are several other methods for solving linear systems, including:

• Graphing: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.
• Elimination: This method involves adding or subtracting the two equations to eliminate one variable.
• Matrices: This method involves using matrices to solve the system.

Q: Which method is the best?

A: The best method depends on the specific problem and your personal preference. Substitution is often the easiest method to use, but graphing and elimination can be useful in certain situations.

Q: Can I use substitution to solve systems with non-linear equations?

A: No, substitution is only useful for solving linear systems. If you have a system with non-linear equations, you may need to use a different method, such as graphing or numerical methods.

Q: Are there any real-world applications of solving linear systems?

A: Yes, there are many real-world applications of solving linear systems, including:

• Economics: Linear systems can be used to model supply and demand curves.
• Engineering: Linear systems can be used to model electrical circuits and mechanical systems.
• Computer Science: Linear systems can be used to solve problems in machine learning and data analysis.

Conclusion

Solving linear systems using substitution is a powerful method that can be used to solve a wide range of problems. By expressing one variable in terms of the other and substituting it into the other equation, we can eliminate one variable and solve for the other. This method has many real-world applications and is an essential skill for students and professionals alike.