Is The Point { (6,7)$}$ A Solution To The System Shown?${ \begin{align*} y &= 2x + 1 \ 5x - 3y &= 9 \end{align*} }$A. No, Because The Provided Point Makes The Equation ${ 5x - 3y = 9\$} True, But Not The Equation [$y
Understanding the System of Equations
A system of equations is a set of two or more equations that contain the same variables. In this case, we have two equations:
- y = 2x + 1
- 5x - 3y = 9
To determine if the point (6,7) is a solution to the system, we need to substitute the values of x and y into both equations and check if they are true.
Substituting the Values into the First Equation
Let's start by substituting the values of x and y into the first equation:
y = 2x + 1
- x = 6
- y = 7
Substituting these values into the equation, we get:
7 = 2(6) + 1
Expanding the equation, we get:
7 = 12 + 1
Simplifying the equation, we get:
7 = 13
This equation is not true, as 7 is not equal to 13. Therefore, the point (6,7) does not satisfy the first equation.
Substituting the Values into the Second Equation
Now, let's substitute the values of x and y into the second equation:
5x - 3y = 9
- x = 6
- y = 7
Substituting these values into the equation, we get:
5(6) - 3(7) = 9
Expanding the equation, we get:
30 - 21 = 9
Simplifying the equation, we get:
9 = 9
This equation is true, as 9 is equal to 9. Therefore, the point (6,7) satisfies the second equation.
Conclusion
Based on the analysis, we can conclude that the point (6,7) is not a solution to the system of equations. Although it satisfies the second equation, it does not satisfy the first equation. Therefore, the correct answer is:
A. No, because the provided point makes the equation 5x - 3y = 9 true, but not the equation y = 2x + 1
Why is the Point Not a Solution?
The point (6,7) is not a solution to the system because it does not satisfy both equations. In the first equation, the point does not satisfy the equation y = 2x + 1, as 7 is not equal to 2(6) + 1. In the second equation, the point satisfies the equation 5x - 3y = 9, as 5(6) - 3(7) = 9. However, this is not enough to make the point a solution to the system, as it must satisfy both equations.
What is a Solution to the System?
A solution to the system of equations is a point that satisfies both equations. In other words, it is a point that makes both equations true. To find a solution to the system, we need to find a point that satisfies both equations. This can be done by solving the system of equations using various methods, such as substitution or elimination.
How to Solve the System of Equations
There are several methods to solve a system of equations, including:
- Substitution Method: This method involves substituting the values of x and y into one of the equations and solving for the other variable.
- Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
- Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Conclusion
In conclusion, the point (6,7) is not a solution to the system of equations. Although it satisfies the second equation, it does not satisfy the first equation. Therefore, the correct answer is:
A. No, because the provided point makes the equation 5x - 3y = 9 true, but not the equation y = 2x + 1
Final Thoughts
Solving systems of equations is an important concept in mathematics, as it has numerous applications in various fields, such as physics, engineering, and economics. By understanding how to solve systems of equations, we can analyze and solve complex problems that involve multiple variables.
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that contain the same variables. In other words, it is a collection of equations that must be solved simultaneously to find the values of the variables.
Q: How do I know if a point is a solution to a system of equations?
A: To determine if a point is a solution to a system of equations, you need to substitute the values of x and y into both equations and check if they are true. If the point satisfies both equations, then it is a solution to the system.
Q: What is the difference between a solution and a point of intersection?
A: A solution to a system of equations is a point that satisfies both equations. A point of intersection, on the other hand, is a point that lies on both graphs of the equations. Not all points of intersection are solutions to the system, and not all solutions are points of intersection.
Q: How do I solve a system of equations?
A: There are several methods to solve a system of equations, including:
- Substitution Method: This method involves substituting the values of x and y into one of the equations and solving for the other variable.
- Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
- Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: What is the substitution method?
A: The substitution method is a method of solving a system of equations by substituting the values of x and y into one of the equations and solving for the other variable. This method is useful when one of the equations is easily solvable.
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 are the same.
Q: What is the graphical method?
A: The graphical method is a method of solving a system of equations by graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear.
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. Then, find the point of intersection of the two graphs.
Q: What is the point of intersection?
A: The point of intersection is a point that lies on both graphs of the equations. It is the solution to the system of equations.
Q: How do I find the point of intersection?
A: To find the point of intersection, you need to graph the equations on a coordinate plane and find the point where the two graphs intersect.
Q: What is the significance of solving systems of equations?
A: Solving systems of equations is an important concept in mathematics, as it has numerous applications in various fields, such as physics, engineering, and economics. By understanding how to solve systems of equations, we can analyze and solve complex problems that involve multiple variables.
Q: What are some real-world applications of solving systems of equations?
A: Some real-world applications of solving systems of equations include:
- Physics: Solving systems of equations is used to describe the motion of objects in physics.
- Engineering: Solving systems of equations is used to design and optimize systems in engineering.
- Economics: Solving systems of equations is used to model and analyze economic systems.
Q: How do I practice solving systems of equations?
A: To practice solving systems of equations, you can try the following:
- Practice solving systems of equations with different methods: Try solving systems of equations using different methods, such as substitution, elimination, and graphical methods.
- Use online resources: There are many online resources available that provide practice problems and exercises for solving systems of equations.
- Work with a partner: Working with a partner can help you stay motivated and learn from each other.
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: Make sure to check the solution to ensure that it satisfies both equations.
- Not using the correct method: Choose the correct method for solving the system of equations.
- Not graphing the equations correctly: Make sure to graph the equations correctly on a coordinate plane.
Q: How do I know if I have made a mistake when solving a system of equations?
A: If you have made a mistake when solving a system of equations, you may notice that the solution does not satisfy both equations. You can also check your work by graphing the equations on a coordinate plane and finding the point of intersection.