Find The Solution Of This System Of Equations. Separate The \[$x\$\]- And \[$y\$\]-values With A Comma.$\[ \begin{aligned} x &= 0 + Y \\ 3x + 12y &= 15 \end{aligned} \\]Enter The Correct Answer.
Introduction
In mathematics, a system of linear equations is a set of two or more equations that are linear in nature. These equations are used to represent a relationship between variables, and solving them is an essential skill in mathematics and other fields. In this article, we will focus on solving a system of two linear equations with two variables, x and y. We will use a step-by-step approach to find the solution of the system of equations.
Understanding the System of Equations
The system of equations we will be solving is given by:
{ \begin{aligned} x &= 0 + y \\ 3x + 12y &= 15 \end{aligned} \}
This system consists of two linear equations with two variables, x and y. The first equation is a simple linear equation, while the second equation is a linear equation with a coefficient of 3 for x and 12 for y.
Solving the System of Equations
To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the method of substitution.
Step 1: Solve the First Equation for x
The first equation is:
We can simplify this equation by combining the constants:
This equation tells us that x is equal to y.
Step 2: Substitute x into the Second Equation
Now that we have an expression for x in terms of y, we can substitute it into the second equation:
Substituting x = y into this equation, we get:
Step 3: Simplify the Equation
Combining like terms, we get:
Step 4: Solve for y
Dividing both sides of the equation by 15, we get:
Step 5: Find the Value of x
Now that we have the value of y, we can find the value of x by substituting y into the equation x = y:
Step 6: Write the Solution as an Ordered Pair
The solution to the system of equations is the ordered pair (x, y) = (1, 1).
Conclusion
In this article, we solved a system of two linear equations with two variables, x and y. We used the method of substitution to find the solution of the system of equations. The solution is the ordered pair (x, y) = (1, 1). This solution satisfies both equations in the system.
Tips and Tricks
- When solving a system of linear equations, it's essential to check your work by plugging the solution back into both equations.
- If you're having trouble solving a system of equations, try using the method of elimination or substitution.
- Make sure to simplify your equations as much as possible to make it easier to solve them.
Real-World Applications
Solving systems of linear equations has many real-world applications, including:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
- Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
Final Thoughts
Solving systems of linear equations is an essential skill in mathematics and other fields. By following the steps outlined in this article, you can solve systems of linear equations with ease. Remember to check your work, simplify your equations, and use the method of substitution or elimination to find the solution. With practice, you'll become proficient in solving systems of linear equations and be able to apply this skill to real-world problems.
Introduction
Solving systems of linear equations is a fundamental concept in mathematics and other fields. In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables, x and y. In this article, we will address some of the most frequently asked questions related to solving systems of linear equations.
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more equations that are linear in nature. These equations are used to represent a relationship between variables, and solving them is an essential skill in mathematics and other fields.
Q: What are the different methods for solving systems of linear equations?
A: There are two main methods for solving systems of linear equations: the method of substitution and the method of elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.
Q: How do I know which method to use?
A: The choice of method depends on the specific system of equations. If the equations are easy to solve, the method of substitution may be the best choice. If the equations are more complex, the method of elimination may be more effective.
Q: What if I have a system of three or more equations?
A: If you have a system of three or more equations, you can use the method of substitution or elimination to solve the system. However, you may need to use a more advanced technique, such as Gaussian elimination or matrix operations.
Q: Can I use a calculator to solve systems of linear equations?
A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "solve" function.
Q: How do I check my work when solving a system of linear equations?
A: To check your work, plug the solution back into both equations to make sure it satisfies both equations. If the solution does not satisfy both equations, you may need to re-evaluate your work.
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 your work
- Not simplifying the equations
- Not using the correct method for the specific system of equations
- Not plugging the solution back into both equations to check your work
Q: How do I apply the concept of solving systems of linear equations to real-world problems?
A: Solving systems of linear equations has many real-world applications, including physics, engineering, computer science, and economics. To apply the concept to real-world problems, you need to identify the variables and equations involved, and then use the methods of substitution or elimination to solve the system.
Q: What are some advanced topics related to solving systems of linear equations?
A: Some advanced topics related to solving systems of linear equations include:
- Gaussian elimination
- Matrix operations
- Determinants
- Inverse matrices
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics and other fields. By understanding the different methods for solving systems of linear equations, you can apply the concept to real-world problems and become proficient in solving systems of linear equations.
Tips and Tricks
- Always check your work by plugging the solution back into both equations.
- Simplify the equations as much as possible to make it easier to solve them.
- Use the correct method for the specific system of equations.
- Practice, practice, practice! The more you practice solving systems of linear equations, the more comfortable you will become with the concept.
Real-World Applications
Solving systems of linear equations has many real-world applications, including:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
- Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
- Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
Final Thoughts
Solving systems of linear equations is an essential skill in mathematics and other fields. By understanding the different methods for solving systems of linear equations, you can apply the concept to real-world problems and become proficient in solving systems of linear equations. Remember to check your work, simplify the equations, and use the correct method for the specific system of equations. With practice, you'll become proficient in solving systems of linear equations and be able to apply this skill to real-world problems.