Solve The System By The Addition Method.${ \begin{array}{l} 3x + 2y = 9 \ 3x - 2y = 9 \end{array} }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Solution Set Is
Introduction
The addition method is a technique used to solve systems of linear equations. It involves adding two or more equations together to eliminate one of the variables. In this article, we will use the addition method to solve a system of two linear equations.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} 3x + 2y = 9 \\ 3x - 2y = 9 \end{array} \}
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.
Step 1: Add the Two Equations Together
To solve the system using the addition method, we will add the two equations together. This will eliminate one of the variables.
{ \begin{array}{l} (3x + 2y) + (3x - 2y) = 9 + 9 \\ 6x = 18 \end{array} \}
By adding the two equations together, we have eliminated the variable y. We are now left with an equation that contains only the variable x.
Step 2: Solve for x
Now that we have eliminated the variable y, we can solve for x. To do this, we will divide both sides of the equation by 6.
{ \frac{6x}{6} = \frac{18}{6} \\ x = 3 \end{array} \}
Therefore, the value of x is 3.
Step 3: Substitute x into One of the Original Equations
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We will use the first equation:
{ 3x + 2y = 9 \end{array} \}
Substituting x = 3 into this equation, we get:
{ 3(3) + 2y = 9 \\ 9 + 2y = 9 \end{array} \}
Step 4: Solve for y
Now that we have substituted x into the equation, we can solve for y. To do this, we will subtract 9 from both sides of the equation.
{ 9 + 2y - 9 = 9 - 9 \\ 2y = 0 \end{array} \}
Dividing both sides of the equation by 2, we get:
{ \frac{2y}{2} = \frac{0}{2} \\ y = 0 \end{array} \}
Therefore, the value of y is 0.
Conclusion
Using the addition method, we have solved the system of equations:
{ \begin{array}{l} 3x + 2y = 9 \\ 3x - 2y = 9 \end{array} \}
The solution to the system is x = 3 and y = 0.
Discussion
The addition method is a useful technique for solving systems of linear equations. It involves adding two or more equations together to eliminate one of the variables. In this article, we used the addition method to solve a system of two linear equations. We found that the solution to the system is x = 3 and y = 0.
Example
Here is an example of how to use the addition method to solve a system of linear equations:
{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} \}
To solve this system, we will add the two equations together:
{ (2x + 3y) + (x - 2y) = 12 + (-3) \\ 3x + y = 9 \end{array} \}
We can then solve for x and y using the same steps as before.
Applications
The addition method has many applications in mathematics and science. It is used to solve systems of linear equations that arise in a variety of fields, including physics, engineering, and economics.
Summary
In this article, we used the addition method to solve a system of two linear equations. We found that the solution to the system is x = 3 and y = 0. The addition method is a useful technique for solving systems of linear equations, and it has many applications in mathematics and science.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Jim Hefferon
- [3] "Linear Algebra: A Modern Introduction" by David Poole
Keywords
- Addition method
- System of linear equations
- Linear algebra
- Mathematics
- Science
- Physics
- Engineering
- Economics
Solve the System by the Addition Method: Q&A =====================================================
Introduction
In our previous article, we used the addition method to solve a system of two linear equations. In this article, we will answer some frequently asked questions about the addition method and provide additional examples to help you understand the concept better.
Q: What is the addition method?
A: The addition method is a technique used to solve systems of linear equations. It involves adding two or more equations together to eliminate one of the variables.
Q: How do I know which equations to add together?
A: To determine which equations to add together, look for the variables that you want to eliminate. Add the equations together in such a way that the variable you want to eliminate is eliminated.
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 addition method to eliminate one variable at a time. Start by eliminating one variable, then use the resulting equation to eliminate another variable, and so on.
Q: Can I use the addition method to solve a system of nonlinear equations?
A: No, the addition method is only used to solve systems of linear equations. If you have a system of nonlinear equations, you will need to use a different method, such as substitution or elimination.
Q: How do I know if the addition method will work for a particular system of equations?
A: The addition method will work for a system of equations if the coefficients of the variables are the same in both equations. If the coefficients are not the same, you will need to use a different method.
Q: Can I use the addition method to solve a system of equations with fractions?
A: Yes, you can use the addition method to solve a system of equations with fractions. Just be sure to multiply both sides of the equation by the least common multiple of the denominators to eliminate the fractions.
Q: What if I get a contradiction when using the addition method?
A: If you get a contradiction when using the addition method, it means that the system of equations has no solution. This can happen if the equations are inconsistent or if the coefficients are not the same.
Q: Can I use the addition method to solve a system of equations with decimals?
A: Yes, you can use the addition method to solve a system of equations with decimals. Just be sure to round the coefficients and constants to the nearest hundredth to avoid any errors.
Q: How do I know if the solution to a system of equations is unique?
A: The solution to a system of equations is unique if the equations are consistent and the coefficients are not the same. If the equations are inconsistent or the coefficients are the same, the solution may not be unique.
Q: Can I use the addition method to solve a system of equations with absolute values?
A: No, the addition method is not used to solve systems of equations with absolute values. If you have a system of equations with absolute values, you will need to use a different method, such as substitution or elimination.
Q: How do I know if the addition method is the best method to use for a particular system of equations?
A: The addition method is the best method to use for a system of equations if the coefficients of the variables are the same in both equations. If the coefficients are not the same, you may need to use a different method, such as substitution or elimination.
Example 1
Solve the system of equations using the addition method:
{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} \}
To solve this system, we will add the two equations together:
{ (2x + 3y) + (x - 2y) = 12 + (-3) \\ 3x + y = 9 \end{array} \}
We can then solve for x and y using the same steps as before.
Example 2
Solve the system of equations using the addition method:
{ \begin{array}{l} x + 2y = 6 \\ 2x - 3y = -3 \end{array} \}
To solve this system, we will add the two equations together:
{ (x + 2y) + (2x - 3y) = 6 + (-3) \\ 3x - y = 3 \end{array} \}
We can then solve for x and y using the same steps as before.
Conclusion
In this article, we have answered some frequently asked questions about the addition method and provided additional examples to help you understand the concept better. The addition method is a useful technique for solving systems of linear equations, and it has many applications in mathematics and science.
Discussion
The addition method is a simple and effective technique for solving systems of linear equations. It involves adding two or more equations together to eliminate one of the variables. In this article, we have shown how to use the addition method to solve systems of equations with fractions, decimals, and absolute values.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Jim Hefferon
- [3] "Linear Algebra: A Modern Introduction" by David Poole
Keywords
- Addition method
- System of linear equations
- Linear algebra
- Mathematics
- Science
- Physics
- Engineering
- Economics