$\[ \begin{array}{l} 2x + 3y = 15 \\ x + Y = 6 \end{array} \\]Work Out The Values Of \[$x\$\] And \[$y\$\].
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Introduction
In mathematics, a system of linear equations is a set of two or more linear 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 with two variables, x and y. We will use the given system of equations:
Our goal is to find the values of x and y that satisfy both equations.
Understanding the System of Equations
A system of linear equations can be represented graphically as a set of lines on a coordinate plane. Each equation is a line, and the solution to the system is the point of intersection between the two lines. In this case, we have two equations:
Method 1: Substitution Method
One way to solve a system of linear equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve Equation 2 for y
We can solve Equation 2 for y by subtracting x from both sides:
Step 2: Substitute the expression for y into Equation 1
Now, we can substitute the expression for y into Equation 1:
Step 3: Simplify the equation
Expanding and simplifying the equation, we get:
Combine like terms:
Subtract 18 from both sides:
Multiply both sides by -1:
Step 4: Find the value of y
Now that we have the value of x, we can find the value of y by substituting x into the expression for y:
Method 2: Elimination Method
Another way to solve a system of linear equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.
Step 1: Multiply Equation 1 by 1 and Equation 2 by -2
To eliminate the variable x, we can multiply Equation 1 by 1 and Equation 2 by -2:
Step 2: Add the two equations
Now, we can add the two equations to eliminate the variable x:
Step 3: Find the value of x
Now that we have the value of y, we can find the value of x by substituting y into one of the original equations:
Subtract 3 from both sides:
Conclusion
In this article, we have solved a system of two linear equations with two variables, x and y. We have used two methods: the substitution method and the elimination method. Both methods have led to the same solution: x = 3 and y = 3. This solution satisfies both equations and represents the point of intersection between the two lines.
Final Answer
The final answer is:
- x = 3
- y = 3
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Introduction
In our previous article, we solved a system of two linear equations with two variables, x and y. We used two methods: the substitution method and the elimination method. In this article, we will provide a Q&A guide to help you understand the concepts and methods used to solve a system of linear equations.
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.
Q: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if the lines represented by the equations intersect at a single point. If the lines are parallel, the system has no solution.
Q: What is the substitution method?
A: The substitution method is a method used to solve a system of linear equations by solving one equation for one variable and then substituting that expression into the other equation.
Q: What is the elimination method?
A: The elimination method is a method used to solve a system of linear equations by adding or subtracting the equations to eliminate one variable.
Q: How do I choose between the substitution method and the elimination method?
A: You can choose between the substitution method and the elimination method based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, the elimination method is usually easier to use. If the coefficients of one variable are different in both equations, the substitution method is usually easier to use.
Q: What if I have a system of linear equations with three variables?
A: If you have a system of linear equations with three variables, you can use the same methods as before: the substitution method and the elimination method. However, you may need to use a combination of both methods to solve the system.
Q: Can I use a graphing calculator to solve a system of linear equations?
A: Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can help you visualize the lines represented by the equations and find the point of intersection.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, it means that the lines represented by the equations are parallel and do not intersect. In this case, the system has no solution.
Q: What if I have a system of linear equations with infinitely many solutions?
A: If you have a system of linear equations with infinitely many solutions, it means that the lines represented by the equations are the same line. In this case, the system has infinitely many solutions.
Q: Can I use a system of linear equations to model real-world problems?
A: Yes, you can use a system of linear equations to model real-world problems. Systems of linear equations can be used to represent relationships between variables in a variety of fields, including business, economics, and science.
Q: How do I know if a system of linear equations is consistent or inconsistent?
A: A system of linear equations is consistent if it has a solution. A system of linear equations is inconsistent if it has no solution.
Q: What is the difference between a consistent and an inconsistent system of linear equations?
A: A consistent system of linear equations has a solution, while an inconsistent system of linear equations has no solution.
Q: Can I use a system of linear equations to solve a problem with multiple variables?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables. Systems of linear equations can be used to represent relationships between multiple variables in a variety of fields.
Q: How do I know if a system of linear equations is a linear combination of the variables?
A: A system of linear equations is a linear combination of the variables if the coefficients of the variables are constants.
Q: What is the importance of solving a system of linear equations?
A: Solving a system of linear equations is important because it can help you understand relationships between variables in a variety of fields, including business, economics, and science.
Q: Can I use a system of linear equations to solve a problem with multiple equations?
A: Yes, you can use a system of linear equations to solve a problem with multiple equations. Systems of linear equations can be used to represent relationships between multiple variables in a variety of fields.
Q: How do I know if a system of linear equations is a system of linear equations with two variables?
A: A system of linear equations is a system of linear equations with two variables if it has two equations and two variables.
Q: What is the difference between a system of linear equations and a system of nonlinear equations?
A: A system of linear equations is a system of equations where the variables are raised to the power of 1, while a system of nonlinear equations is a system of equations where the variables are raised to a power other than 1.
Q: Can I use a system of linear equations to solve a problem with multiple variables and multiple equations?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables and multiple equations. Systems of linear equations can be used to represent relationships between multiple variables in a variety of fields.
Q: How do I know if a system of linear equations is a system of linear equations with three variables?
A: A system of linear equations is a system of linear equations with three variables if it has three equations and three variables.
Q: What is the importance of solving a system of linear equations in real-world problems?
A: Solving a system of linear equations is important in real-world problems because it can help you understand relationships between variables in a variety of fields, including business, economics, and science.
Q: Can I use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context. Systems of linear equations can be used to represent relationships between multiple variables in a variety of fields.
Q: How do I know if a system of linear equations is a system of linear equations with multiple variables and multiple equations?
A: A system of linear equations is a system of linear equations with multiple variables and multiple equations if it has multiple equations and multiple variables.
Q: What is the difference between a system of linear equations and a system of nonlinear equations in a real-world context?
A: A system of linear equations is a system of equations where the variables are raised to the power of 1, while a system of nonlinear equations is a system of equations where the variables are raised to a power other than 1.
Q: Can I use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context. Systems of linear equations can be used to represent relationships between multiple variables in a variety of fields.
Q: How do I know if a system of linear equations is a system of linear equations with multiple variables and multiple equations in a real-world context?
A: A system of linear equations is a system of linear equations with multiple variables and multiple equations in a real-world context if it has multiple equations and multiple variables.
Q: What is the importance of solving a system of linear equations in a real-world context?
A: Solving a system of linear equations is important in a real-world context because it can help you understand relationships between variables in a variety of fields, including business, economics, and science.
Q: Can I use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context. Systems of linear equations can be used to represent relationships between multiple variables in a variety of fields.
Q: How do I know if a system of linear equations is a system of linear equations with multiple variables and multiple equations in a real-world context?
A: A system of linear equations is a system of linear equations with multiple variables and multiple equations in a real-world context if it has multiple equations and multiple variables.
Q: What is the difference between a system of linear equations and a system of nonlinear equations in a real-world context?
A: A system of linear equations is a system of equations where the variables are raised to the power of 1, while a system of nonlinear equations is a system of equations where the variables are raised to a power other than 1.
Q: Can I use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables and multiple equations in a real-world context. Systems of linear equations can be used to represent relationships between multiple variables in a variety of fields.
Q: How do I know if a system of linear equations is a system of linear equations with multiple variables and multiple equations in a real-world context?
A: A system of linear equations is a system of linear equations with multiple variables and multiple equations in a real-world context if it has multiple equations and multiple variables.