Solve The System Of Equations:${ \begin{cases} 2x + Y = 8 \ x = 5 \end{cases} }$

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of equations that consists of two linear equations in two variables. We will use the given system of equations as an example to demonstrate the steps involved in solving it.

The System of Equations

The given system of equations is:

{ \begin{cases} 2x + y = 8 \\ x = 5 \end{cases} \}

This system consists of two linear equations in two variables, x and y. The first equation is 2x + y = 8, and the second equation is x = 5.

Substitution Method

To solve this system of equations, we can use the substitution method. This method involves substituting the value of one variable from one equation into the other equation. In this case, we can substitute the value of x from the second equation into the first equation.

Step 1: Substitute the value of x into the first equation

We are given that x = 5. We can substitute this value into the first equation:

2(5) + y = 8

Step 2: Simplify the equation

Now, we can simplify the equation by multiplying 2 and 5:

10 + y = 8

Step 3: Solve for y

Next, we can solve for y by subtracting 10 from both sides of the equation:

y = -2

Conclusion

In this article, we solved a system of equations using the substitution method. We substituted the value of x from the second equation into the first equation and simplified it to solve for y. The final answer is y = -2.

Example Problems

Problem 1

Solve the system of equations:

{ \begin{cases} x + 2y = 6 \\ x = 3 \end{cases} \}

Solution

We can substitute the value of x from the second equation into the first equation:

(3) + 2y = 6

Next, we can simplify the equation by multiplying 2 and 3:

3 + 2y = 6

Now, we can solve for y by subtracting 3 from both sides of the equation:

2y = 3

Finally, we can divide both sides of the equation by 2 to solve for y:

y = 3/2

Problem 2

Solve the system of equations:

{ \begin{cases} 2x + y = 10 \\ x = 4 \end{cases} \}

Solution

We can substitute the value of x from the second equation into the first equation:

2(4) + y = 10

Next, we can simplify the equation by multiplying 2 and 4:

8 + y = 10

Now, we can solve for y by subtracting 8 from both sides of the equation:

y = 2

Tips and Tricks

Use the Substitution Method

The substitution method is a powerful tool for solving systems of equations. It involves substituting the value of one variable from one equation into the other equation. This method is particularly useful when one of the equations is already solved for one variable.

Check Your Work

When solving a system of equations, it's essential to check your work to ensure that the solution satisfies both equations. You can do this by plugging the values of x and y back into both equations and verifying that they are true.

Conclusion

In this article, we solved a system of equations using the substitution method. We substituted the value of x from the second equation into the first equation and simplified it to solve for y. We also provided example problems and tips and tricks for solving systems of equations. With practice and patience, you can become proficient in solving systems of equations and apply this skill to a wide range of mathematical problems.

Further Reading

If you're interested in learning more about solving systems of equations, I recommend checking out the following resources:

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

These resources provide a comprehensive introduction to solving systems of equations and offer a wealth of practice problems and examples to help you master this skill.

Final Thoughts

Solving systems of equations is an essential skill in mathematics that has numerous applications in science, engineering, economics, and other fields. By mastering this skill, you can solve a wide range of mathematical problems and apply your knowledge to real-world scenarios. With practice and patience, you can become proficient in solving systems of equations and unlock a world of mathematical possibilities.

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Q: What is a system of equations?

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

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

There are several methods for solving systems of equations, including:

  • Substitution method
  • Elimination method
  • Graphical method
  • Matrix method

Q: What is the substitution method?

The substitution method involves substituting the value of one variable from one equation into the other equation.

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the graphical method?

The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the matrix method?

The matrix method involves using matrices to solve the system of equations.

Q: How do I choose which method to use?

The choice of method depends on the type of equations and the variables involved. For example, if the equations are linear and have two variables, the substitution or elimination method may be the best choice.

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

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

  • Not checking the solution to ensure it satisfies both equations
  • Not using the correct method for the type of equations
  • Not simplifying the equations before solving
  • Not checking for extraneous solutions

Q: How do I check my solution to ensure it satisfies both equations?

To check your solution, plug the values of the variables back into both equations and verify that they are true.

Q: What is an extraneous solution?

An extraneous solution is a solution that satisfies one of the equations but not the other.

Q: How do I avoid extraneous solutions?

To avoid extraneous solutions, make sure to check your solution to ensure it satisfies both equations.

Q: Can I use technology to solve systems of equations?

Yes, technology such as calculators and computer software can be used to solve systems of equations.

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

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

  • Physics and engineering
  • Economics and finance
  • Computer science and programming
  • Data analysis and statistics

Q: How can I practice solving systems of equations?

You can practice solving systems of equations by working through example problems and exercises, and by using online resources and software.

Q: What are some common types of systems of equations?

Some common types of systems of equations include:

  • Linear systems
  • Nonlinear systems
  • Systems with two variables
  • Systems with three or more variables

Q: How do I know which type of system of equations I am dealing with?

To determine which type of system of equations you are dealing with, look at the equations and determine if they are linear or nonlinear, and if they have two or more variables.

Q: Can I use the same method for all types of systems of equations?

No, different methods may be required for different types of systems of equations.

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

Some common mistakes to avoid when working with systems of equations include:

  • Not checking the solution to ensure it satisfies both equations
  • Not using the correct method for the type of equations
  • Not simplifying the equations before solving
  • Not checking for extraneous solutions

Q: How can I improve my skills in solving systems of equations?

You can improve your skills in solving systems of equations by practicing regularly, working through example problems and exercises, and using online resources and software.

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

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

  • Khan Academy: Systems of Equations
  • Mathway: Systems of Equations
  • Wolfram Alpha: Systems of Equations
  • Online textbooks and tutorials
  • Practice problems and exercises