Is ( 9 , 6 (9, 6 ( 9 , 6 ] A Solution To This System Of Equations? Y = − X − 1 Y = X − 3 \begin{array}{l} Y = -x - 1 \\ Y = X - 3 \end{array} Y = − X − 1 Y = X − 3 ​ A. Yes B. No

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Introduction

When dealing with systems of equations, it's essential to determine whether a given point satisfies both equations. In this case, we're presented with a system of two linear equations and asked to verify if the point $(9, 6)$ is a solution. To do this, we'll substitute the coordinates of the point into each equation and check if the resulting statement is true.

Understanding the System of Equations

The given system consists of two linear equations:

1. $y = -x - 1$
2. $y = x - 3$

These equations represent lines in the Cartesian coordinate system. To find the solution, we need to determine if the point $(9, 6)$ lies on both lines.

Substituting the Point into the First Equation

Let's start by substituting the coordinates of the point $(9, 6)$ into the first equation:

$y = -x - 1$

We replace $x$ with $9$ and $y$ with $6$:

$6 = -(9) - 1$

Simplifying the equation, we get:

$6 = -9 - 1$

$6 = -10$

This statement is false, as $6$ is not equal to $-10$. Therefore, the point $(9, 6)$ does not satisfy the first equation.

Substituting the Point into the Second Equation

Now, let's substitute the coordinates of the point $(9, 6)$ into the second equation:

$y = x - 3$

We replace $x$ with $9$ and $y$ with $6$:

$6 = 9 - 3$

Simplifying the equation, we get:

$6 = 6$

This statement is true, as $6$ is indeed equal to $6$. Therefore, the point $(9, 6)$ satisfies the second equation.

Conclusion

Based on our analysis, we've found that the point $(9, 6)$ satisfies the second equation but not the first equation. Since a solution to a system of equations must satisfy both equations, the point $(9, 6)$ is not a solution to this system.

The final answer is: B. No

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Introduction

In our previous article, we explored whether the point $(9, 6)$ is a solution to the given system of equations. We found that the point satisfies the second equation but not the first equation. In this Q&A article, we'll delve deeper into the topic and address some common questions related to systems of equations and solutions.

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

A: A solution to a system of equations is a point that satisfies all the equations in the system. In other words, it's a point that makes each equation true.

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, you need to substitute the coordinates of the point into each equation and check if the resulting statement is true. If the point satisfies all the equations, it's a solution.

Q: What if a point satisfies one equation but not the other?

A: If a point satisfies one equation but not the other, it's not a solution to the system. In our case, the point $(9, 6)$ satisfies the second equation but not the first equation, so it's not a solution.

Q: Can a system of equations have multiple solutions?

A: Yes, a system of equations can have multiple solutions. However, it's also possible for a system to have no solutions or an infinite number of solutions.

Q: How do I know if a system of equations has no solutions or an infinite number of solutions?

A: To determine if a system has no solutions or an infinite number of solutions, you need to examine the equations and look for any inconsistencies or contradictions. If the equations are inconsistent, the system has no solutions. If the equations are consistent but dependent, the system has an infinite number of solutions.

Q: What is the difference between a dependent and an independent system of equations?

A: A dependent system of equations has at least one equation that can be expressed as a linear combination of the other equations. In other words, one equation is a multiple of the other equation. An independent system of equations, on the other hand, has no dependent equations.

Q: Can a system of equations have both dependent and independent equations?

A: Yes, a system of equations can have both dependent and independent equations. In this case, the system is said to be inconsistent.

Q: How do I graph a system of equations?

A: To graph a system of equations, you need to plot the lines represented by each equation on the same coordinate plane. The point of intersection of the two lines represents the solution to the system.

Q: What is the significance of the point of intersection in a system of equations?

A: The point of intersection represents the solution to the system. It's the point that satisfies both equations.

Q: Can a system of equations have multiple points of intersection?

A: Yes, a system of equations can have multiple points of intersection. However, in this case, the system would have multiple solutions.

Q: How do I determine if a system of equations has a unique solution?

A: To determine if a system of equations has a unique solution, you need to examine the equations and look for any inconsistencies or contradictions. If the equations are consistent and independent, the system has a unique solution.

Conclusion

In this Q&A article, we've addressed some common questions related to systems of equations and solutions. We've explored the concept of a solution, how to determine if a point is a solution, and the significance of the point of intersection. We've also discussed the differences between dependent and independent systems of equations and how to graph a system of equations.