For Each Ordered Pair, Determine Whether It Is A Solution To The System Of Equations.$\[ \left\{ \begin{array}{c} y = 5x - 3 \\ 15x - 3y = 9 \end{array} \right. \\]$\[ \begin{array}{|c|c|c|} \hline (x, Y) & \text{Yes} & \text{No}

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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 two linear equations in two variables. We will use the method of substitution and elimination to determine whether each ordered pair is a solution to the system of equations.

The System of Equations

The given system of equations is:

{ \left\{ \begin{array}{c} y = 5x - 3 \\ 15x - 3y = 9 \end{array} \right. \}

This system consists of two linear equations in two variables, x and y. The first equation is y = 5x - 3, and the second equation is 15x - 3y = 9.

Method of Substitution

To solve this system of equations, we can use the method of substitution. This method involves substituting the expression for one variable from one equation into the other equation.

Let's start by solving the first equation for y:

y = 5x - 3

Now, substitute this expression for y into the second equation:

15x - 3(5x - 3) = 9

Expand and simplify the equation:

15x - 15x + 9 = 9

Combine like terms:

9 = 9

This equation is true for all values of x. Therefore, the solution to the system of equations is all real numbers.

Method of Elimination

Another method to solve this system of equations is the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables.

Let's start by multiplying the first equation by 3 to make the coefficients of y in both equations the same:

3y = 15x - 9

Now, add this equation to the second equation:

15x - 3y + 3y = 9 + 15x - 9

Combine like terms:

15x = 15x

This equation is true for all values of x. Therefore, the solution to the system of equations is all real numbers.

Checking the Solutions

To check the solutions, we need to plug the values of x and y into both equations and make sure they are true.

Let's take the ordered pair (x, y) = (1, 4). Plug these values into both equations:

Equation 1: y = 5x - 3 4 = 5(1) - 3 4 = 2 This equation is not true.

Equation 2: 15x - 3y = 9 15(1) - 3(4) = 9 15 - 12 = 9 3 = 9 This equation is not true.

Therefore, the ordered pair (x, y) = (1, 4) is not a solution to the system of equations.

Conclusion

In this article, we solved a system of two linear equations in two variables using the method of substitution and elimination. We found that the solution to the system of equations is all real numbers. We also checked the solutions by plugging the values of x and y into both equations and found that the ordered pair (x, y) = (1, 4) is not a solution to the system of equations.

Discussion

The method of substitution and elimination are two common methods used to solve systems of equations. The method of substitution involves substituting the expression for one variable from one equation into the other equation, while the method of elimination involves adding or subtracting the equations to eliminate one of the variables.

The system of equations we solved in this article is a simple example of a system of linear equations in two variables. However, there are many other types of systems of equations, including systems of nonlinear equations, systems of equations with more than two variables, and systems of equations with complex coefficients.

Final Thoughts

Solving systems of equations is an important skill in mathematics and is used in many real-world applications, such as physics, engineering, and economics. By understanding the methods of substitution and elimination, we can solve systems of equations and find the values of the variables.

References

  • [1] "Systems of Equations" by Math Open Reference
  • [2] "Solving Systems of Equations" by Khan Academy
  • [3] "Systems of Linear Equations" by Wolfram MathWorld

Glossary

  • System of equations: A set of two or more equations that are solved simultaneously to find the values of the variables.
  • Linear equation: An equation in which the highest power of the variable is 1.
  • Nonlinear equation: An equation in which the highest power of the variable is greater than 1.
  • Substitution method: A method of solving systems of equations by substituting the expression for one variable from one equation into the other equation.
  • Elimination method: A method of solving systems of equations by adding or subtracting the equations to eliminate one of the variables.
    Frequently Asked Questions: Systems of Equations =====================================================

Q: What is a system of equations?

A: 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 types of systems of equations?

A: There are several types of systems of equations, including:

  • Linear systems of equations: Systems of equations in which the highest power of the variable is 1.
  • Nonlinear systems of equations: Systems of equations in which the highest power of the variable is greater than 1.
  • Systems of equations with more than two variables: Systems of equations in which there are more than two variables.
  • Systems of equations with complex coefficients: Systems of equations in which the coefficients are complex numbers.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including:

  • Substitution method: A method of solving systems of equations by substituting the expression for one variable from one equation into the other equation.
  • Elimination method: A method of solving systems of equations by adding or subtracting the equations to eliminate one of the variables.
  • Graphical method: A method of solving systems of equations by 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 of solving systems of equations by substituting the expression for one variable from one equation into the other equation.

Q: What is the elimination method?

A: The elimination method is a method of solving systems of equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I check the solutions to a system of equations?

A: To check the solutions to a system of equations, plug the values of the variables into both equations and make sure they are true.

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 solutions: Make sure to check the solutions to the system of equations to ensure they are true.
  • Not using the correct method: Choose the correct method to solve the system of equations, such as substitution or elimination.
  • Not simplifying the equations: Simplify the equations before solving the system of equations.

Q: How do I apply systems of equations in real-world situations?

A: Systems of equations are used in many real-world situations, including:

  • Physics: Systems of equations are used to describe the motion of objects and the forces acting on them.
  • Engineering: Systems of equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Systems of equations are used to model economic systems and make predictions about economic trends.

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

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

  • Textbooks: There are many textbooks available that cover systems of equations in detail.
  • Online resources: Websites such as Khan Academy and Math Open Reference offer video lessons and interactive exercises on systems of equations.
  • Tutorials: Many online tutorials and courses are available that cover systems of equations in detail.

Q: How do I practice solving systems of equations?

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

  • Practice problems: Work on practice problems to reinforce your understanding of systems of equations.
  • Online exercises: Use online resources such as Khan Academy and Math Open Reference to practice solving systems of equations.
  • Real-world applications: Apply systems of equations to real-world situations to see how they are used in practice.