Determine If The Point \[$(2, -2)\$\] Is The Point Of Intersection For The Given Lines. In Other Words, Is \[$(2, -2)\$\] The Solution To The System Of Equations?$\[ \begin{array}{r} -7x + 8y = -30 \\ 5x - 7y =

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. One of the key concepts in solving systems of equations is the point of intersection, which is the point where the two lines represented by the equations intersect. In this article, we will determine if the point {(2, -2)$}$ is the point of intersection for the given lines, which are represented by the system of equations:

−7x+8y=−305x−7y=10\begin{array}{r} -7x + 8y = -30 \\ 5x - 7y = 10 \end{array}

Understanding the System of Equations

To determine if the point {(2, -2)$}$ is the point of intersection, we need to understand the system of equations. The first equation is:

−7x+8y=−30-7x + 8y = -30

This equation represents a line in the coordinate plane, where the x-coordinate is multiplied by -7 and the y-coordinate is multiplied by 8. The constant term -30 is the y-intercept of the line.

The second equation is:

5x−7y=105x - 7y = 10

This equation also represents a line in the coordinate plane, where the x-coordinate is multiplied by 5 and the y-coordinate is multiplied by -7. The constant term 10 is the y-intercept of the line.

Substituting the Point into the Equations

To determine if the point {(2, -2)$}$ is the point of intersection, we need to substitute the x and y values into both equations and check if the resulting equations are true.

For the first equation, we substitute x = 2 and y = -2:

−7(2)+8(−2)=−14−16=−30-7(2) + 8(-2) = -14 - 16 = -30

This equation is true, as the left-hand side equals the right-hand side.

For the second equation, we substitute x = 2 and y = -2:

5(2)−7(−2)=10+14=245(2) - 7(-2) = 10 + 14 = 24

This equation is not true, as the left-hand side does not equal the right-hand side.

Conclusion

Based on the substitution of the point {(2, -2)$}$ into both equations, we can conclude that the point is not the point of intersection for the given lines. The first equation is satisfied, but the second equation is not.

Why the Point is Not the Point of Intersection

The point {(2, -2)$}$ is not the point of intersection because it does not satisfy both equations simultaneously. The first equation is satisfied, but the second equation is not. This means that the point lies on one of the lines, but not on the other line.

Implications of the Result

The result of this analysis has implications for various fields, including mathematics, physics, and engineering. In mathematics, it highlights the importance of checking the validity of solutions to systems of equations. In physics and engineering, it can affect the accuracy of calculations and the design of systems.

Real-World Applications

The concept of point of intersection has numerous real-world applications, including:

  • Computer Graphics: In computer graphics, the point of intersection is used to determine the intersection of two lines or curves.
  • Physics: In physics, the point of intersection is used to determine the collision of two objects.
  • Engineering: In engineering, the point of intersection is used to determine the intersection of two curves or surfaces.

Conclusion

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

A: The point of intersection is the point where two or more lines represented by the equations intersect. It is the solution to the system of equations.

Q: How do I determine if a point is the point of intersection?

A: To determine if a point is the point of intersection, you need to substitute the x and y values into both equations and check if the resulting equations are true.

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

A: If the point satisfies one equation but not the other, it means that the point lies on one of the lines, but not on the other line. Therefore, it is not the point of intersection.

Q: Can a point be the point of intersection if it satisfies both equations?

A: Yes, a point can be the point of intersection if it satisfies both equations. This means that the point lies on both lines and is the solution to the system of equations.

Q: How do I find the point of intersection if I have two equations?

A: To find the point of intersection, you can use the following steps:

  1. Write down the two equations.
  2. Substitute the x and y values into both equations.
  3. Check if the resulting equations are true.
  4. If the point satisfies both equations, it is the point of intersection.

Q: What if I have a system of three or more equations?

A: If you have a system of three or more equations, you can use the same steps as above to find the point of intersection. However, you may need to use more advanced techniques, such as substitution or elimination, to solve the system of equations.

Q: Can I use a graphing calculator to find the point of intersection?

A: Yes, you can use a graphing calculator to find the point of intersection. Simply graph the two equations on the calculator and find the point where they intersect.

Q: What are some real-world applications of the point of intersection?

A: The point of intersection has numerous real-world applications, including:

  • Computer Graphics: In computer graphics, the point of intersection is used to determine the intersection of two lines or curves.
  • Physics: In physics, the point of intersection is used to determine the collision of two objects.
  • Engineering: In engineering, the point of intersection is used to determine the intersection of two curves or surfaces.

Q: How do I check if a point is the point of intersection in a system of equations?

A: To check if a point is the point of intersection, you need to substitute the x and y values into both equations and check if the resulting equations are true. If the point satisfies both equations, it is the point of intersection.

Q: What if I have a system of equations with fractions or decimals?

A: If you have a system of equations with fractions or decimals, you can use the same steps as above to find the point of intersection. However, you may need to use more advanced techniques, such as substitution or elimination, to solve the system of equations.

Q: Can I use a computer program to find the point of intersection?

A: Yes, you can use a computer program to find the point of intersection. Simply input the two equations into the program and it will output the point of intersection.

Conclusion

In conclusion, the point of intersection is an important concept in mathematics that has numerous real-world applications. By following the steps outlined above, you can determine if a point is the point of intersection in a system of equations.