Solve The System Of Equations:${ \begin{cases} 9x - 3y = 9 \ 3x - Y = 3 \end{cases} }$
**Solve the System of Equations: A Step-by-Step Guide** =====================================================
Introduction
Solving a system 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 focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.
The System of Equations
The given system of equations is:
{ \begin{cases} 9x - 3y = 9 \\ 3x - y = 3 \end{cases} \}
What is a System of Equations?
A system of equations is a set of two or more equations that involve the same variables. In this case, we have two equations with two variables, x and y. The goal is to find the values of x and y that satisfy both equations simultaneously.
Why is Solving a System of Equations Important?
Solving a system of equations is important in many real-world applications, such as:
- Physics: To solve problems involving motion, forces, and energies.
- Engineering: To design and optimize systems, such as electrical circuits and mechanical systems.
- Economics: To model and analyze economic systems, such as supply and demand.
- Computer Science: To solve problems involving algorithms and data structures.
How to Solve a System of Equations
There are several methods to solve a system of equations, including:
- Substitution Method: Substitute one equation into the other to eliminate one variable.
- Elimination Method: Add or subtract the equations to eliminate one variable.
- Graphical Method: Plot the equations on a graph and find the intersection point.
Step-by-Step Solution
In this article, we will use the substitution method to solve the given system of equations.
Step 1: Write Down the Equations
Write down the given system of equations:
{ \begin{cases} 9x - 3y = 9 \\ 3x - y = 3 \end{cases} \}
Step 2: Solve One Equation for One Variable
Solve the second equation for y:
Step 3: Substitute the Expression into the Other Equation
Substitute the expression for y into the first equation:
Step 4: Simplify the Equation
Simplify the equation:
Step 5: Check the Solution
Check the solution by plugging the values back into both equations:
The solution satisfies both equations.
Conclusion
Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we used the substitution method to solve a system of two linear equations with two variables. We provided a step-by-step guide on how to solve the system of equations and highlighted the importance of solving systems of equations in various real-world applications.
Frequently Asked Questions
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.
Q: Why is solving a system of equations important?
A: Solving a system of equations is important in many real-world applications, such as physics, engineering, economics, and computer science.
Q: What are the methods to solve a system of equations?
A: There are several methods to solve a system of equations, including substitution, elimination, and graphical methods.
Q: How do I choose the method to solve a system of equations?
A: Choose the method that is most suitable for the given system of equations. For example, use the substitution method if one equation is already solved for one variable.
Q: What if I get stuck while solving a system of equations?
A: Don't worry! Take a break, come back to the problem later, and try a different method or approach. You can also ask for help from a teacher, tutor, or online resource.
Additional Resources
- Khan Academy: Systems of Equations
- Mathway: Systems of Equations
- Wolfram Alpha: Systems of Equations
Final Thoughts
Solving a system of equations is a fundamental concept in mathematics that requires practice and patience. With this article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables. We also highlighted the importance of solving systems of equations in various real-world applications. Remember to choose the method that is most suitable for the given system of equations and don't be afraid to ask for help if you get stuck. Happy solving!