Solve The Following System Of Equations:$\[ \begin{cases} 5x + 3y = 15 \\ x - 6y = 3 \end{cases} \\]
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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.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are the variables.
Example of a System of Linear Equations
The following is an example of a system of two linear equations with two variables:
{ \begin{cases} 5x + 3y = 15 \\ x - 6y = 3 \end{cases} \}
This system consists of two linear equations, each involving the variables x and y.
Methods for Solving a System of Linear Equations
There are several methods for solving a system of linear equations, including:
- Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
- Elimination Method: This method involves adding or subtracting the equations in the system to eliminate one of the variables.
- Graphical Method: This method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.
Solving the System of Linear Equations using the Substitution Method
To solve the system of linear equations using the substitution method, we will first solve one equation for one variable. Let's solve the second equation for x:
x = 3 + 6y
Now, we will substitute this expression for x into the first equation:
5(3 + 6y) + 3y = 15
Expanding and simplifying the equation, we get:
15 + 30y + 3y = 15
Combine like terms:
33y = 0
Divide both sides by 33:
y = 0
Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the second equation:
x - 6(0) = 3
Simplify the equation:
x = 3
Therefore, the solution to the system of linear equations is x = 3 and y = 0.
Solving the System of Linear Equations using the Elimination Method
To solve the system of linear equations using the elimination method, we will add the two equations in the system to eliminate one of the variables. Let's add the two equations:
(5x + 3y) + (x - 6y) = 15 + 3
Combine like terms:
6x - 3y = 18
Now, we will multiply the second equation by 3 to make the coefficients of y in both equations opposite:
3(x - 6y) = 3(3)
Simplify the equation:
3x - 18y = 9
Now, we will add the two equations:
(6x - 3y) + (3x - 18y) = 18 + 9
Combine like terms:
9x - 21y = 27
Now, we will divide both sides by 9:
x - 7/3y = 3
Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the first equation:
5x + 3y = 15
Substitute x = 3:
5(3) + 3y = 15
Simplify the equation:
15 + 3y = 15
Subtract 15 from both sides:
3y = 0
Divide both sides by 3:
y = 0
Therefore, the solution to the system of linear equations is x = 3 and y = 0.
Conclusion
Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. There are several methods for solving a system of linear equations, including the substitution method and the elimination method. In this article, we have used both methods to solve a system of two linear equations with two variables. The solution to the system is x = 3 and y = 0.
Final Answer
The final answer is x = 3 and y = 0.
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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. Each equation in the system is a linear equation, which means that it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are 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 two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.
Q: What are the different methods for solving a system of linear equations?
A: There are several methods for solving a system of linear equations, including:
- Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
- Elimination Method: This method involves adding or subtracting the equations in the system to eliminate one of the variables.
- Graphical Method: This method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.
Q: How do I choose which method to use?
A: The choice of method depends on the specific system of linear equations and the variables involved. If the system has two variables, the substitution method and elimination method are often the most effective. If the system has more than two variables, the graphical method may be more useful.
Q: What if I have a system of linear equations with more than two variables?
A: If you have a system of linear equations with more than two variables, you can use the graphical method to find the solution. This involves graphing the equations in the system on a coordinate plane and finding the point of intersection.
Q: Can I use a calculator to solve a system of linear equations?
A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "solve" function.
Q: What if I make a mistake while solving a system of linear equations?
A: If you make a mistake while solving a system of linear equations, you can try to identify the error and correct it. If you are unable to correct the error, you may need to start over from the beginning.
Q: Can I use a computer program to solve a system of linear equations?
A: Yes, you can use a computer program to solve a system of linear equations. Many computer programs, such as MATLAB and Python, have built-in functions for solving systems of linear equations.
Q: How do I know if a system of linear equations is consistent or inconsistent?
A: A system of linear equations is consistent if the two equations do not contradict each other. If the two equations contradict each other, then the system is inconsistent.
Q: What is the difference between a consistent and 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 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 model a wide range of problems, including problems involving finance, science, and engineering.
Q: How do I apply a system of linear equations to a real-world problem?
A: To apply a system of linear equations to a real-world problem, you need to identify the variables involved and the relationships between them. You can then use the system of linear equations to model the problem and find the solution.
Q: What are some common applications of systems of linear equations?
A: Some common applications of systems of linear equations include:
- Finance: Systems of linear equations can be used to model financial problems, such as investment portfolios and loan payments.
- Science: Systems of linear equations can be used to model scientific problems, such as population growth and chemical reactions.
- Engineering: Systems of linear equations can be used to model engineering problems, such as bridge design and circuit analysis.
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 model problems with multiple variables, such as finance, science, and engineering problems.
Q: How do I know if a system of linear equations has a unique solution?
A: A system of linear equations has a unique solution if the two equations are consistent and the system has only one solution. If the system has multiple solutions, then it is not unique.
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, then the system is inconsistent. This means that the two equations contradict each other and there is no solution.
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 model problems with multiple equations, such as finance, science, and engineering problems.
Q: How do I know if a system of linear equations is dependent or independent?
A: A system of linear equations is dependent if the two equations are identical. If the two equations are not identical, then the system is independent.
Q: What is the difference between a dependent and independent system of linear equations?
A: A dependent system of linear equations has multiple solutions, while an independent system of linear equations has a unique solution.
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 model problems with multiple variables and multiple equations, such as finance, science, and engineering problems.
Q: How do I apply a system of linear equations to a problem with multiple variables and multiple equations?
A: To apply a system of linear equations to a problem with multiple variables and multiple equations, you need to identify the variables involved and the relationships between them. You can then use the system of linear equations to model the problem and find the solution.
Q: What are some common applications of systems of linear equations with multiple variables and multiple equations?
A: Some common applications of systems of linear equations with multiple variables and multiple equations include:
- Finance: Systems of linear equations can be used to model financial problems, such as investment portfolios and loan payments.
- Science: Systems of linear equations can be used to model scientific problems, such as population growth and chemical reactions.
- Engineering: Systems of linear equations can be used to model engineering problems, such as bridge design and circuit analysis.
Q: Can I use a system of linear equations to solve a problem with multiple variables, multiple equations, and no solution?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables, multiple equations, and no solution. Systems of linear equations can be used to model problems with multiple variables, multiple equations, and no solution, such as finance, science, and engineering problems.
Q: How do I know if a system of linear equations with multiple variables, multiple equations, and no solution is consistent or inconsistent?
A: A system of linear equations with multiple variables, multiple equations, and no solution is inconsistent if the two equations contradict each other. If the two equations do not contradict each other, then the system is consistent.
Q: What is the difference between a consistent and inconsistent system of linear equations with multiple variables, multiple equations, and no solution?
A: A consistent system of linear equations with multiple variables, multiple equations, and no solution has no solution, while an inconsistent system of linear equations with multiple variables, multiple equations, and no solution has a solution.
Q: Can I use a system of linear equations to solve a problem with multiple variables, multiple equations, and multiple solutions?
A: Yes, you can use a system of linear equations to solve a problem with multiple variables, multiple equations, and multiple solutions. Systems of linear equations can be used to model problems with multiple variables, multiple equations, and multiple solutions, such as finance, science, and engineering problems.
Q: How do I know if a system of linear equations with multiple variables, multiple equations, and multiple solutions is dependent or independent?
A: A system of linear equations with multiple variables, multiple equations, and multiple solutions is dependent if the two equations are identical. If the two equations are not identical, then the system is independent.
Q: What is the difference between a dependent and independent system of linear equations with multiple variables, multiple equations, and multiple solutions?
A: A dependent system of linear equations with multiple variables, multiple equations, and multiple solutions has multiple solutions, while an independent system of linear equations with multiple variables, multiple equations, and multiple solutions has a unique solution.