For Each Ordered Pair, Determine Whether It Is A Solution To The System Of Equations.$\[ \begin{cases} 2x + 5y = -9 \\ y = 4x + 7 \end{cases} \\]$\[ \begin{array}{|c|c|c|} \hline (x, Y) & \text{Yes} & \text{No} \\ \hline (-2, -1) & 0 &
Introduction
Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore how to determine whether a given ordered pair is a solution to a system of equations. We will use a specific system of equations as an example and analyze each ordered pair to determine whether it is a solution.
The System of Equations
The system of equations we will be working with is:
{ \begin{cases} 2x + 5y = -9 \\ y = 4x + 7 \end{cases} \}
This system consists of two linear equations in two variables, x and y. The first equation is 2x + 5y = -9, and the second equation is y = 4x + 7.
Ordered Pairs
To determine whether an ordered pair is a solution to the system of equations, we need to substitute the values of x and y into both equations and check if the resulting statements are true.
Substitution Method
The substitution method involves substituting the value of y from the second equation into the first equation. This will give us an equation in one variable, which we can then solve to find the value of x.
Analyzing the Ordered Pairs
We will analyze each ordered pair to determine whether it is a solution to the system of equations.
Ordered Pair 1: (-2, -1)
To determine whether the ordered pair (-2, -1) is a solution to the system of equations, we need to substitute the values of x and y into both equations.
Step 1: Substitute x = -2 and y = -1 into the first equation
2x + 5y = -9 2(-2) + 5(-1) = -9 -4 - 5 = -9 -9 = -9
Step 2: Substitute x = -2 and y = -1 into the second equation
y = 4x + 7 -1 = 4(-2) + 7 -1 = -8 + 7 -1 = -1
Since both equations are true, the ordered pair (-2, -1) is a solution to the system of equations.
Conclusion
In this article, we analyzed each ordered pair to determine whether it is a solution to the system of equations. We used the substitution method to substitute the values of x and y into both equations and checked if the resulting statements are true. We found that the ordered pair (-2, -1) is a solution to the system of equations.
Discussion
Solving systems of equations is an important concept in mathematics that has numerous applications in various fields, including science, engineering, and economics. The substitution method is a powerful tool for solving systems of equations, and it can be used to find the values of variables that satisfy multiple equations simultaneously.
Future Work
In future work, we can explore other methods for solving systems of equations, such as the elimination method and the graphing method. We can also analyze more complex systems of equations and explore their applications in various fields.
References
- [1] "Solving Systems of Equations" by Math Open Reference
- [2] "Systems of Equations" by Khan Academy
Appendix
The following is a list of ordered pairs that we analyzed in this article:
| (x, y) | Yes | No |
|---|---|---|
| (-2, -1) |
Note: The "Yes" column indicates that the ordered pair is a solution to the system of equations, and the "No" column indicates that it is not a solution.
Introduction
Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will answer some of the most frequently asked questions about solving systems of equations.
Q: What is a system of equations?
A system of equations is a set of two or more equations that involve the same variables. For example:
{ \begin{cases} 2x + 5y = -9 \\ y = 4x + 7 \end{cases} \}
This system consists of two linear equations in two variables, x and y.
Q: How do I solve a system of equations?
There are several methods for solving systems of equations, including the substitution method, the elimination method, and the graphing method. The substitution method involves substituting the value of y from one equation into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.
Q: What is the substitution method?
The substitution method involves substituting the value of y from one equation into the other equation. For example, if we have the system:
{ \begin{cases} 2x + 5y = -9 \\ y = 4x + 7 \end{cases} \}
We can substitute the value of y from the second equation into the first equation:
2x + 5(4x + 7) = -9
Q: What is the elimination method?
The elimination method involves adding or subtracting the equations to eliminate one of the variables. For example, if we have the system:
{ \begin{cases} 2x + 5y = -9 \\ 3x + 2y = 5 \end{cases} \}
We can add the two equations to eliminate the y-variable:
(2x + 5y) + (3x + 2y) = -9 + 5 5x + 7y = -4
Q: How do I know which method to use?
The choice of method depends on the type of equations and the variables involved. If the equations are linear and involve two variables, the substitution method or the elimination method may be used. If the equations are non-linear or involve more than two variables, the graphing method may be used.
Q: What is the graphing method?
The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. For example, if we have the system:
{ \begin{cases} y = 2x - 3 \\ y = x + 2 \end{cases} \}
We can graph the two equations on a coordinate plane and find the point of intersection.
Q: How do I graph a system of equations?
To graph a system of equations, we need to graph each equation separately and find the point of intersection. We can use a graphing calculator or graph paper to graph the equations.
Q: What is the point of intersection?
The point of intersection is the point where the two graphs meet. This point represents the solution to the system of equations.
Q: How do I find the point of intersection?
To find the point of intersection, we need to graph the two equations separately and find the point where they meet. We can use a graphing calculator or graph paper to graph the equations.
Q: What if the graphs do not intersect?
If the graphs do not intersect, it means that the system of equations has no solution.
Q: What if the graphs intersect at multiple points?
If the graphs intersect at multiple points, it means that the system of equations has multiple solutions.
Conclusion
Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we answered some of the most frequently asked questions about solving systems of equations.
Discussion
Solving systems of equations is an important concept in mathematics that has numerous applications in various fields, including science, engineering, and economics. The substitution method, the elimination method, and the graphing method are all useful tools for solving systems of equations.
Future Work
In future work, we can explore other methods for solving systems of equations, such as the matrix method and the Gaussian elimination method. We can also analyze more complex systems of equations and explore their applications in various fields.
References
- [1] "Solving Systems of Equations" by Math Open Reference
- [2] "Systems of Equations" by Khan Academy
Appendix
The following is a list of frequently asked questions about solving systems of equations:
| Q | A |
|---|---|
| What is a system of equations? | A set of two or more equations that involve the same variables. |
| How do I solve a system of equations? | There are several methods, including the substitution method, the elimination method, and the graphing method. |
| What is the substitution method? | The substitution method involves substituting the value of y from one equation into the other equation. |
| What is the elimination method? | The elimination method involves adding or subtracting the equations to eliminate one of the variables. |
| How do I know which method to use? | The choice of method depends on the type of equations and the variables involved. |
| What is the graphing method? | The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. |
| How do I graph a system of equations? | We need to graph each equation separately and find the point of intersection. |
| What is the point of intersection? | The point of intersection is the point where the two graphs meet. |
| How do I find the point of intersection? | We need to graph the two equations separately and find the point where they meet. |
| What if the graphs do not intersect? | If the graphs do not intersect, it means that the system of equations has no solution. |
| What if the graphs intersect at multiple points? | If the graphs intersect at multiple points, it means that the system of equations has multiple solutions. |