Which Of The Following Systems Of Equations Has The Solution \[$(1,6)\$\]?A. $\[ \begin{array}{l} y = -5x - 1 \\ y = -x + 7 \end{array} \\]B. $\[ \begin{array}{l} y = 5x + 1 \\ y = X - 7 \end{array} \\]C.
Introduction
When dealing with systems of equations, it's essential to determine which system has a specific solution. In this case, we're given the solution {(1,6)$}$ and need to find out which of the following systems of equations has this solution.
System A
The first system of equations is given by:
{ \begin{array}{l} y = -5x - 1 \\ y = -x + 7 \end{array} \}
To determine if this system has the solution {(1,6)$}$, we need to substitute the values of x and y into both equations and check if they are true.
Substituting x = 1 and y = 6 into the first equation
Substituting x = 1 and y = 6 into the first equation, we get:
Simplifying the equation, we get:
This is not true, so the first equation does not have the solution {(1,6)$}$.
Substituting x = 1 and y = 6 into the second equation
Substituting x = 1 and y = 6 into the second equation, we get:
Simplifying the equation, we get:
This is true, so the second equation has the solution {(1,6)$}$.
System B
The second system of equations is given by:
{ \begin{array}{l} y = 5x + 1 \\ y = x - 7 \end{array} \}
To determine if this system has the solution {(1,6)$}$, we need to substitute the values of x and y into both equations and check if they are true.
Substituting x = 1 and y = 6 into the first equation
Substituting x = 1 and y = 6 into the first equation, we get:
Simplifying the equation, we get:
This is true, so the first equation has the solution {(1,6)$}$.
Substituting x = 1 and y = 6 into the second equation
Substituting x = 1 and y = 6 into the second equation, we get:
Simplifying the equation, we get:
This is not true, so the second equation does not have the solution {(1,6)$}$.
Conclusion
Based on the analysis, we can conclude that:
- System A has the solution {(1,6)$}$ only in the second equation.
- System B has the solution {(1,6)$}$ only in the first equation.
Therefore, the correct answer is that System B has the solution {(1,6)$}$.
Discussion
This problem requires the student to analyze the given systems of equations and determine which one has the solution {(1,6)$}$. The student needs to substitute the values of x and y into both equations and check if they are true. This problem helps the student to develop their problem-solving skills and to understand the concept of systems of equations.
Tips and Tricks
- When dealing with systems of equations, it's essential to substitute the values of x and y into both equations and check if they are true.
- Use the given solution to determine which system has the solution.
- Analyze each equation separately and check if it has the solution.
- Use the concept of substitution to solve the problem.
Example Problems
- Find the solution to the system of equations:
{ \begin{array}{l} y = 2x + 1 \\ y = x - 3 \end{array} \}
- Find the solution to the system of equations:
{ \begin{array}{l} y = x + 2 \\ y = 2x - 1 \end{array} \}
Key Concepts
- Systems of equations
- Substitution method
- Problem-solving skills
References
- [1] "Systems of Equations" by Math Open Reference
- [2] "Substitution Method" by Khan Academy
Related Topics
- Systems of Equations
- Substitution Method
- Problem-Solving Skills
Introduction
Systems of equations are a fundamental concept in mathematics, and they can be a bit tricky to understand at first. However, with practice and patience, you can master the art of solving systems of equations. In this article, we'll answer some frequently asked questions (FAQs) on systems of equations to help you better understand this concept.
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that are related to each other. Each equation in the system is a statement that two expressions are equal, and the system is a collection of these statements.
Q: How do I solve a system of equations?
A: There are several methods to solve a system of equations, including the substitution method, the elimination method, and the graphing method. The substitution method involves substituting the expression for one variable 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?
A: The substitution method is a method of solving a system of equations by substituting the expression for one variable from one equation into the other equation. This method is useful when one of the equations is linear and the other equation is quadratic.
Q: What is the elimination method?
A: The elimination method is a method of solving a system of equations by adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of the variables in the two equations are the same.
Q: How do I determine which method to use?
A: To determine which method to use, you need to examine the equations and see if they can be easily solved using the substitution or elimination method. If one of the equations is linear and the other equation is quadratic, the substitution method may be more suitable. If the coefficients of the variables in the two equations are the same, the elimination method may be more suitable.
Q: What is the graphing method?
A: The graphing method is a method of solving a system of equations by graphing the two equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and the system has a unique solution.
Q: How do I graph a system of equations?
A: To graph a system of equations, you need to graph each equation separately on a coordinate plane and find the point of intersection. You can use a graphing calculator or a computer program to graph the equations.
Q: What is the difference between a system of equations and a system of inequalities?
A: A system of inequalities is a set of two or more inequalities that are related to each other. The main difference between a system of equations and a system of inequalities is that a system of equations is a set of equalities, while a system of inequalities is a set of inequalities.
Q: How do I solve a system of inequalities?
A: To solve a system of inequalities, you need to find the region of the coordinate plane that satisfies all the inequalities. You can use a graphing calculator or a computer program to graph the inequalities and find the region.
Q: What is the importance of systems of equations in real-life applications?
A: Systems of equations have many real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
Q: How can I practice solving systems of equations?
A: You can practice solving systems of equations by working on problems and exercises in a textbook or online resource. You can also use a graphing calculator or a computer program to graph the equations and find the solution.
Q: What are some common mistakes to avoid when solving systems of equations?
A: Some common mistakes to avoid when solving systems of equations include:
- Not checking the solution to see if it satisfies both equations
- Not using the correct method to solve the system
- Not checking for extraneous solutions
- Not graphing the equations to visualize the solution
Conclusion
Systems of equations are a fundamental concept in mathematics, and they have many real-life applications. By understanding the different methods of solving systems of equations, you can master the art of solving these types of problems. Remember to practice regularly and avoid common mistakes to become proficient in solving systems of equations.
Tips and Tricks
- Always check the solution to see if it satisfies both equations
- Use the correct method to solve the system
- Check for extraneous solutions
- Graph the equations to visualize the solution
- Practice regularly to become proficient in solving systems of equations
Example Problems
- Solve the system of equations:
{ \begin{array}{l} y = 2x + 1 \\ y = x - 3 \end{array} \}
- Solve the system of inequalities:
{ \begin{array}{l} y > 2x + 1 \\ y < x - 3 \end{array} \}
Key Concepts
- Systems of equations
- Substitution method
- Elimination method
- Graphing method
- Systems of inequalities
References
- [1] "Systems of Equations" by Math Open Reference
- [2] "Substitution Method" by Khan Academy
- [3] "Elimination Method" by Mathway
- [4] "Graphing Method" by Wolfram Alpha
Related Topics
- Systems of Equations
- Substitution Method
- Elimination Method
- Graphing Method
- Systems of Inequalities