Solve The System Of Linear Equations By Substitution.$\[ \begin{array}{l} 2x + 3y = 36 \\ x = -2 + Y \\ x = \\ y = \\ \end{array} \\]

Feb 26, 2025 by ADMIN 134 views

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving a system of linear equations using the substitution method. This method involves solving one equation for a variable and then substituting that expression into the other equation.

The System of Linear Equations

The given system of linear equations is:

{ \begin{array}{l} 2x + 3y = 36 \\ x = -2 + y \\ \end{array} \}

We are given two equations:

$2x + 3y = 36$
$x = -2 + y$

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

Step 1: Solve the Second Equation for x

The second equation is already solved for $x$ , which is:

$x = -2 + y$

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

Step 2: Substitute the Expression for x into the First Equation

Now, we will substitute the expression for $x$ into the first equation:

$2x + 3y = 36$

Substituting $x = -2 + y$ into the first equation, we get:

$2(-2 + y) + 3y = 36$

Step 3: Simplify the Equation

Expanding and simplifying the equation, we get:

$-4 + 2y + 3y = 36$

Combine like terms:

$-4 + 5y = 36$

Step 4: Solve for y

Add 4 to both sides of the equation:

$5y = 40$

Divide both sides by 5:

$y = 8$

Step 5: Find the Value of x

Now that we have the value of $y$ , we can substitute it into the expression for $x$ :

$x = -2 + y$

Substituting $y = 8$ into the expression for $x$ , we get:

$x = -2 + 8$

$x = 6$

Conclusion

In this article, we solved a system of linear equations using the substitution method. We started by solving the second equation for $x$ , and then substituted the expression for $x$ into the first equation. We simplified the equation and solved for $y$ , and then used the value of $y$ to find the value of $x$ . The final solution is:

$x = 6$ $y = 8$

This solution satisfies both equations in the system.

Applications of Solving Systems of Linear Equations

Solving systems of linear equations has numerous applications in various fields such as:

Physics: Solving systems of linear equations is used to describe the motion of objects in terms of their position, velocity, and acceleration.
Engineering: Solving systems of linear equations is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
Economics: Solving systems of linear equations is used to model economic systems and make predictions about the behavior of economic variables.
Computer Science: Solving systems of linear equations is used in computer graphics, game development, and machine learning.

Tips and Tricks for Solving Systems of Linear Equations

Here are some tips and tricks for solving systems of linear equations:

Use the substitution method when one equation is already solved for a variable.
Use the elimination method when the coefficients of the variables are the same.
Use the graphing method when the system of equations is not easily solvable using algebraic methods.
Check your solutions by plugging them back into the original equations.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields. In this article, we solved a system of linear equations using the substitution method, and we provided tips and tricks for solving systems of linear equations. We hope that this article has provided you with a better understanding of how to solve systems of linear equations and how to apply this concept in real-world problems.

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Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: What are the different methods for solving systems of linear equations?

A: There are three main methods for solving systems of linear equations:

Substitution Method: This method involves solving one equation for a variable and then substituting that expression into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
Graphing Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a method for solving systems of linear equations where one equation is already solved for a variable. We substitute the expression for the variable into the other equation and solve for the remaining variable.

Q: What is the elimination method?

A: The elimination method is a method for solving systems of linear equations where the coefficients of the variables are the same. We add or subtract the equations to eliminate one of the variables and solve for the remaining variable.

Q: What is the graphing method?

A: The graphing method is a method for solving systems of linear equations where we graph the equations on a coordinate plane and find the point of intersection.

Q: How do I know which method to use?

A: The choice of method depends on the type of system of equations and the ease of solving it. If one equation is already solved for a variable, use the substitution method. If the coefficients of the variables are the same, use the elimination method. If the system is not easily solvable using algebraic methods, use the graphing method.

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

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

Not checking the solutions by plugging them back into the original equations.
Not using the correct method for the type of system of equations.
Not simplifying the equations before solving.
Not checking for extraneous solutions.

Q: How do I check my solutions?

A: To check your solutions, plug the values back into the original equations and make sure they are true. If the values satisfy both equations, then they are the correct solutions.

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

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

Physics: Solving systems of linear equations is used to describe the motion of objects in terms of their position, velocity, and acceleration.
Engineering: Solving systems of linear equations is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
Economics: Solving systems of linear equations is used to model economic systems and make predictions about the behavior of economic variables.
Computer Science: Solving systems of linear equations is used in computer graphics, game development, and machine learning.

Q: How do I practice solving systems of linear equations?

A: To practice solving systems of linear equations, try the following:

Start with simple systems of equations and gradually move to more complex ones.
Use online resources such as Khan Academy, Mathway, or Wolfram Alpha to practice solving systems of linear equations.
Try solving systems of linear equations with different methods such as substitution, elimination, and graphing.
Practice solving systems of linear equations with different types of equations such as linear, quadratic, and polynomial.

Q: What are some resources for learning more about solving systems of linear equations?

A: Some resources for learning more about solving systems of linear equations include:

Khan Academy: Khan Academy has a comprehensive course on solving systems of linear equations.
Mathway: Mathway is an online resource that can help you solve systems of linear equations.
Wolfram Alpha: Wolfram Alpha is an online resource that can help you solve systems of linear equations.
Online textbooks: There are many online textbooks that provide detailed explanations and examples of solving systems of linear equations.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields. In this article, we answered some frequently asked questions about solving systems of linear equations and provided resources for learning more about this topic. We hope that this article has provided you with a better understanding of how to solve systems of linear equations and how to apply this concept in real-world problems.