Is { (9,8)$}$ A Solution To This System Of Equations?${ \begin{array}{l} 6x - 5y = 16 \ x = 9 \end{array} }$A. Yes B. No

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Understanding the System of Equations

A system of equations is a set of two or more equations that contain multiple variables. In this case, we have a system of two equations with two variables, x and y. The first equation is 6x - 5y = 16, and the second equation is x = 9. To determine if the point (9,8) is a solution to this system of equations, we need to substitute the values of x and y into both equations and check if they are true.

Substituting Values into the First Equation

To substitute the values of x and y into the first equation, we need to replace x with 9 and y with 8. The first equation becomes:

6(9) - 5(8) = 16

Simplifying the Equation

To simplify the equation, we need to multiply 6 by 9 and 5 by 8, and then subtract the two products.

6(9) = 54 5(8) = 40

Now, we can rewrite the equation as:

54 - 40 = 16

Evaluating the Equation

To evaluate the equation, we need to subtract 40 from 54.

54 - 40 = 14

Conclusion

The equation 54 - 40 = 16 is not true, since 54 - 40 = 14. This means that the point (9,8) is not a solution to the first equation.

Substituting Values into the Second Equation

To substitute the values of x into the second equation, we need to replace x with 9.

9 = 9

Evaluating the Equation

To evaluate the equation, we can see that 9 is indeed equal to 9. This means that the point (9,8) is a solution to the second equation.

Conclusion

Since the point (9,8) is not a solution to the first equation, but it is a solution to the second equation, we need to determine if it is a solution to the system of equations.

Determining if the Point is a Solution

A point is a solution to a system of equations if it satisfies both equations. In this case, the point (9,8) satisfies the second equation, but it does not satisfy the first equation. Therefore, the point (9,8) is not a solution to the system of equations.

Answer

The answer is B. No.

Why is the Point Not a Solution?

The point (9,8) is not a solution to the system of equations because it does not satisfy the first equation. The first equation is 6x - 5y = 16, and when we substitute x = 9 and y = 8, we get 54 - 40 = 16, which is not true. This means that the point (9,8) does not satisfy the first equation, and therefore, it is not a solution to the system of equations.

What is a Solution to the System of Equations?

A solution to a system of equations is a point that satisfies both equations. In this case, we need to find a point that satisfies both the first equation 6x - 5y = 16 and the second equation x = 9.

Finding a Solution

To find a solution, we can substitute x = 9 into the first equation and solve for y.

6(9) - 5y = 16

Simplifying the Equation

To simplify the equation, we need to multiply 6 by 9 and then subtract 5y from both sides.

54 - 5y = 16

Adding 5y to Both Sides

To add 5y to both sides, we need to isolate y.

54 = 16 + 5y

Subtracting 16 from Both Sides

To subtract 16 from both sides, we need to isolate y.

38 = 5y

Dividing Both Sides by 5

To divide both sides by 5, we need to isolate y.

y = 38/5

Evaluating the Equation

To evaluate the equation, we can divide 38 by 5.

y = 7.6

Conclusion

The point (9,7.6) is a solution to the system of equations because it satisfies both equations. The first equation is 6x - 5y = 16, and when we substitute x = 9 and y = 7.6, we get 54 - 40 = 16, which is true. The second equation is x = 9, and when we substitute x = 9, we get 9 = 9, which is also true.

Answer

The answer is A. Yes.

Why is the Point a Solution?

The point (9,7.6) is a solution to the system of equations because it satisfies both equations. The first equation is 6x - 5y = 16, and when we substitute x = 9 and y = 7.6, we get 54 - 40 = 16, which is true. This means that the point (9,7.6) satisfies the first equation. The second equation is x = 9, and when we substitute x = 9, we get 9 = 9, which is also true. This means that the point (9,7.6) satisfies the second equation. Therefore, the point (9,7.6) is a solution to the system of equations.

Conclusion

In conclusion, the point (9,8) is not a solution to the system of equations because it does not satisfy the first equation. However, the point (9,7.6) is a solution to the system of equations because it satisfies both equations. Therefore, the answer is B. No, the point (9,8) is not a solution to the system of equations, and the answer is A. Yes, the point (9,7.6) is a solution to the system of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain multiple variables. In this case, we have a system of two equations with two variables, x and y.

Q: How do I determine 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.

Q: What is a solution to a system of equations?

A: A solution to a system of equations is a point that satisfies both equations. In other words, it is a point that makes both equations true.

Q: How do I find a solution to a system of equations?

A: To find a solution to a system of equations, you can use substitution or elimination methods. Substitution involves substituting one equation into the other, while elimination involves adding or subtracting the equations to eliminate one of the variables.

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, while a point of intersection is a point that is common to two or more graphs. Not all points of intersection are solutions to a system of equations.

Q: Can a system of equations have multiple solutions?

A: Yes, a system of equations can have multiple solutions. This occurs when the system has infinitely many solutions, meaning that there are an infinite number of points that satisfy both equations.

Q: Can a system of equations have no solutions?

A: Yes, a system of equations can have no solutions. This occurs when the two equations are inconsistent, meaning that they cannot both be true at the same time.

Q: How do I determine if a system of equations has multiple solutions or no solutions?

A: To determine if a system of equations has multiple solutions or no solutions, you can use the following methods:

  • Graphing: Graph the two equations on a coordinate plane and see if they intersect at a single point, multiple points, or no points.
  • Substitution: Substitute one equation into the other and solve for the variable.
  • Elimination: Add or subtract the equations to eliminate one of the variables and solve for the other variable.

Q: What is the importance of solving systems of equations?

A: Solving systems of equations is important in many real-world applications, such as:

  • Physics: Solving systems of equations is used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Solving systems of equations is used to model economic systems and make predictions about future trends.

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 if the system has multiple solutions or no solutions.
  • Not using the correct method to solve the system (e.g. substitution or elimination).
  • Not checking if the solution satisfies both equations.
  • Not considering the possibility of infinitely many solutions.

Q: How can I practice solving systems of equations?

A: You can practice solving systems of equations by:

  • Working through examples and exercises in a textbook or online resource.
  • Using online tools and calculators to solve systems of equations.
  • Creating your own systems of equations and solving them.
  • Joining a study group or working with a tutor to practice solving systems of equations.

Q: What are some advanced topics in systems of equations?

A: Some advanced topics in systems of equations include:

  • Systems of linear equations with multiple variables.
  • Systems of nonlinear equations.
  • Systems of equations with parameters.
  • Systems of equations with constraints.

Q: How can I apply systems of equations to real-world problems?

A: You can apply systems of equations to real-world problems by:

  • Using systems of equations to model real-world systems, such as electrical circuits and mechanical systems.
  • Using systems of equations to make predictions about future trends, such as economic trends.
  • Using systems of equations to design and optimize systems, such as engineering systems.
  • Using systems of equations to solve problems in physics, such as motion and force problems.