Solve The System Of Equations:${ \begin{aligned} 2x - Y &= 39 \ -x &= 6y \end{aligned} }$
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Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{aligned} 2x - y &= 39 \\ -x &= 6y \end{aligned} \}
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.
Method of Substitution
One of the methods used to solve a system of linear equations is the method of substitution. This method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.
Let's start by solving the second equation for x:
{ -x = 6y \}
Multiplying both sides by -1, we get:
{ x = -6y \}
Now, substitute this expression for x into the first equation:
{ 2(-6y) - y = 39 \}
Simplifying the equation, we get:
{ -12y - y = 39 \}
Combine like terms:
{ -13y = 39 \}
Now, divide both sides by -13:
{ y = -3 \}
Finding the Value of x
Now that we have found the value of y, we can substitute it into one of the original equations to find the value of x. Let's use the second equation:
{ -x = 6y \}
Substitute y = -3:
{ -x = 6(-3) \}
Simplifying the equation, we get:
{ -x = -18 \}
Multiply both sides by -1:
{ x = 18 \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of substitution. We started by solving one of the equations for one of the variables and then substituted that expression into the other equation. We then solved for the value of the other variable and finally found the value of the first variable.
Tips and Tricks
- When solving a system of linear equations, it's essential to check your work by plugging the values of the variables back into the original equations to ensure that they are true.
- If you're having trouble solving a system of linear equations, try using a different method, such as the method of elimination or the method of matrices.
- When solving a system of linear equations, it's essential to be careful with the signs of the coefficients and the variables.
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 flow of fluids.
- 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 and make predictions about the behavior of markets.
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. By using the method of substitution, we can solve systems of linear equations and find the values of the variables that satisfy all the equations in the system. With practice and patience, you can become proficient in solving systems of linear equations and apply this skill to a wide range of 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 involve the same set of variables. In other words, it's a collection of equations that can be solved simultaneously to find the values of the variables.
Q: What are the different methods for solving systems of linear equations?
A: There are several methods for solving systems of linear equations, including:
- Method of Substitution: This method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.
- Method of Elimination: This method involves adding or subtracting the equations to eliminate one of the variables.
- Method of Matrices: This method involves using matrices to represent the system of equations and then solving for the values of the variables.
Q: How do I choose the method of substitution?
A: To choose the method of substitution, you need to identify which variable to solve for first. You can do this by looking at the coefficients of the variables in the equations. If one of the coefficients is 1, it's usually easier to solve for that variable first.
Q: What if I get stuck while solving a system of linear equations?
A: If you get stuck while solving a system of linear equations, try the following:
- Check your work: Make sure you've solved the equations correctly and that you haven't made any mistakes.
- Use a different method: Try using a different method, such as the method of elimination or the method of matrices.
- Ask for help: Don't be afraid to ask for help from a teacher, tutor, or classmate.
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: What are some real-world applications of solving systems of linear equations?
A: 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 flow of fluids.
- 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 and make predictions about the behavior of markets.
Q: Can I use systems of linear equations to solve problems in other areas of mathematics?
A: Yes, you can use systems of linear equations to solve problems in other areas of mathematics, such as:
- Algebra: Systems of linear equations can be used to solve quadratic equations and other types of equations.
- Geometry: Systems of linear equations can be used to find the intersection points of lines and planes.
- Calculus: Systems of linear equations can be used to solve optimization problems and other types of problems.
Q: How can I practice solving systems of linear equations?
A: There are many ways to practice solving systems of linear equations, including:
- Solving problems: Try solving problems from a textbook or online resource.
- Using online tools: Use online tools, such as calculators or software, to practice solving systems of linear equations.
- Working with a partner: Work with a partner or classmate to practice solving systems of linear equations.
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the different methods for solving systems of linear equations and practicing with problems, you can become proficient in solving these types of equations and apply this skill to a wide range of real-world problems.