Is $(-1, -2)$ A Solution To This System Of Equations?${ \begin{array}{l} 12x - 2y = 13 \ 14x - 4y = -6 \end{array} }$A. Yes B. No

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Understanding the Problem

To determine if the point $(-1, -2)$ is a solution to the given system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the resulting statement is true. The system of equations is:

{ \begin{array}{l} 12x - 2y = 13 \\ 14x - 4y = -6 \end{array} \}

Substituting the Values

We will substitute $x = -1$ and $y = -2$ into each equation and simplify.

Equation 1: 12x - 2y = 13

Substituting the values of $x$ and $y$ into the first equation, we get:

12(−1)−2(−2)=1312(-1) - 2(-2) = 13

Simplifying the equation, we get:

−12+4=13-12 + 4 = 13

−8=13-8 = 13

This statement is false.

Equation 2: 14x - 4y = -6

Substituting the values of $x$ and $y$ into the second equation, we get:

14(−1)−4(−2)=−614(-1) - 4(-2) = -6

Simplifying the equation, we get:

−14+8=−6-14 + 8 = -6

−6=−6-6 = -6

This statement is true.

Conclusion

Since the statement in Equation 1 is false, we can conclude that the point $(-1, -2)$ is not a solution to the given system of equations.

Why is this the case?

When we substitute the values of $x$ and $y$ into each equation, we get two statements. If both statements are true, then the point is a solution to the system of equations. However, if one or both statements are false, then the point is not a solution to the system of equations.

In this case, the statement in Equation 1 is false, which means that the point $(-1, -2)$ is not a solution to the system of equations.

What does this mean?

This means that the point $(-1, -2)$ does not satisfy both equations simultaneously. In other words, it does not lie on both lines at the same time.

What are the implications of this?

The implications of this are that the point $(-1, -2)$ is not a solution to the system of equations. This means that we cannot use this point as a solution to the system of equations.

How can we use this information?

We can use this information to determine if a point is a solution to a system of equations. We can substitute the values of $x$ and $y$ into each equation and check if the resulting statement is true. If both statements are true, then the point is a solution to the system of equations. If one or both statements are false, then the point is not a solution to the system of equations.

What are some real-world applications of this?

Some real-world applications of this include:

  • Graphing systems of equations: We can use this information to determine if a point lies on a line.
  • Solving systems of equations: We can use this information to determine if a point is a solution to a system of equations.
  • Analyzing data: We can use this information to determine if a point is a solution to a system of equations, which can be useful in data analysis.

Conclusion

In conclusion, the point $(-1, -2)$ is not a solution to the given system of equations. This means that it does not satisfy both equations simultaneously. We can use this information to determine if a point is a solution to a system of equations, which can be useful in graphing systems of equations, solving systems of equations, and analyzing data.

Final Thoughts

In this article, we discussed whether the point $(-1, -2)$ is a solution to the given system of equations. We substituted the values of $x$ and $y$ into each equation and simplified. We found that the statement in Equation 1 is false, which means that the point $(-1, -2)$ is not a solution to the system of equations. We also discussed some real-world applications of this, including graphing systems of equations, solving systems of equations, and analyzing data.

Frequently Asked Questions

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 is a point that lies on all the lines at the same time.

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 each equation and check if the resulting statement is true. If both statements are true, then the point is a solution to the system of equations.

Q: What if one or both statements are false?

A: If one or both statements are false, then the point is not a solution to the system of equations.

Q: Why is this important?

A: This is important because it helps us determine if a point lies on a line or not. It is also useful in solving systems of equations and analyzing data.

Q: Can you give an example of how to use this information?

A: Let's say we have a system of equations:

{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}

We want to determine if the point $(2, 1)$ is a solution to this system of equations. We substitute the values of $x$ and $y$ into each equation and simplify:

2(2)+3(1)=72(2) + 3(1) = 7

4+3=74 + 3 = 7

7=77 = 7

This statement is true.

x−2(1)=−3x - 2(1) = -3

2−2=−32 - 2 = -3

0=−30 = -3

This statement is false.

Since one of the statements is false, we can conclude that the point $(2, 1)$ is not a solution to the system of equations.

Q: What are some real-world applications of this?

A: Some real-world applications of this include:

  • Graphing systems of equations: We can use this information to determine if a point lies on a line.
  • Solving systems of equations: We can use this information to determine if a point is a solution to a system of equations.
  • Analyzing data: We can use this information to determine if a point is a solution to a system of equations, which can be useful in data analysis.

Q: Can you give some tips for solving systems of equations?

A: Here are some tips for solving systems of equations:

  • Read the problem carefully: Make sure you understand what the problem is asking.
  • Substitute the values of $x$ and $y$ into each equation: This will help you determine if the point is a solution to the system of equations.
  • Simplify the equations: This will help you determine if the point is a solution to the system of equations.
  • Check if both statements are true: If both statements are true, then the point is a solution to the system of equations.

Q: What if I get stuck?

A: If you get stuck, don't worry! You can always ask for help or try a different approach. Some common mistakes to avoid include:

  • Not reading the problem carefully: Make sure you understand what the problem is asking.
  • Not substituting the values of $x$ and $y$ into each equation: This is an important step in determining if a point is a solution to a system of equations.
  • Not simplifying the equations: This can make it difficult to determine if the point is a solution to the system of equations.

Q: Can you give some examples of systems of equations?

A: Here are some examples of systems of equations:

  • Linear equations: 2x+3y=72x + 3y = 7 and x−2y=−3x - 2y = -3
  • Quadratic equations: x2+4y2=16x^2 + 4y^2 = 16 and x2−4y2=0x^2 - 4y^2 = 0
  • Polynomial equations: x3+2y3=27x^3 + 2y^3 = 27 and x3−2y3=0x^3 - 2y^3 = 0

Q: Can you give some tips for graphing systems of equations?

A: Here are some tips for graphing systems of equations:

  • Plot the lines: This will help you visualize the system of equations.
  • Find the intersection point: This will help you determine if the point is a solution to the system of equations.
  • Check if the point lies on both lines: If the point lies on both lines, then it is a solution to the system of equations.

Q: What if I'm still having trouble?

A: If you're still having trouble, don't worry! You can always ask for help or try a different approach. Some common mistakes to avoid include:

  • Not plotting the lines: This can make it difficult to visualize the system of equations.
  • Not finding the intersection point: This can make it difficult to determine if the point is a solution to the system of equations.
  • Not checking if the point lies on both lines: This can make it difficult to determine if the point is a solution to the system of equations.