Why Statement A is the Correct Answer

Oct 13, 2025 by recoveryshake.com 113 views

When dealing with equations, it's essential to understand the relationship between points on the graph and the values that make the equation true. In this article, we'll explore the concept of points on the graph of an equation and determine which statement must be true if the point ${$ (1,4)$}$ is on the graph.

What Does it Mean for a Point to be on the Graph of an Equation?

A point ${$ (x,y)$}$ is said to be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ satisfy the equation. In other words, when we substitute the values of ${$ x$}$ and ${$ y$}$ into the equation, the equation holds true.

Analyzing Statement A

Statement A claims that the values ${$ x=1$}$ and ${$ y=4$}$ make the equation true. This statement is true if the point ${$ (1,4)$}$ is on the graph of the equation. When we substitute ${$ x=1$}$ and ${$ y=4$}$ into the equation, the equation should hold true.

Analyzing Statement B

Statement B claims that the values ${$ x=1$}$ and ${$ y=4$}$ are the only values that make the equation true. This statement is not necessarily true. There may be other values of ${$ x$}$ and ${$ y$}$ that satisfy the equation, in addition to ${$ x=1$}$ and ${$ y=4$}$.

Why Statement A is the Correct Answer

The correct answer is Statement A. If the point ${$ (1,4)$}$ is on the graph of an equation, then the values ${$ x=1$}$ and ${$ y=4$}$ must make the equation true. This is because the point ${$ (1,4)$}$ is defined as the values ${$ x=1$}$ and ${$ y=4$}$ that satisfy the equation.

Example: Understanding the Relationship Between Points and Equations

Let's consider an example to illustrate the concept. Suppose we have the equation ${$ x^2 + y^2 = 25$}$. If the point ${$ (3,4)$}$ is on the graph of this equation, then the values ${$ x=3$}$ and ${$ y=4$}$ must make the equation true.

Substituting ${$ x=3$}$ and ${$ y=4$}$ into the equation, we get:

${$ (3)^2 + (4)^2 = 25$ [$9 + 16 = 25$ [$25 = 25$

As we can see, the equation holds true when we substitute [ $x=3\$}$ and ${$ y=4$}$ into the equation. Therefore, the point ${$ (3,4)$}$ is on the graph of the equation.

Conclusion

In conclusion, if the point ${$ (1,4)$}$ is on the graph of an equation, then the values ${$ x=1$}$ and ${$ y=4$}$ must make the equation true. This is because the point ${$ (1,4)$}$ is defined as the values ${$ x=1$}$ and ${$ y=4$}$ that satisfy the equation.

Frequently Asked Questions

Q: What does it mean for a point to be on the graph of an equation?

A: A point ${$ (x,y)$}$ is said to be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ satisfy the equation.

Q: Why is Statement A the correct answer?

A: Statement A is the correct answer because if the point ${$ (1,4)$}$ is on the graph of an equation, then the values ${$ x=1$}$ and ${$ y=4$}$ must make the equation true.

Q: Can there be other values of ${$ x$}$ and ${$ y$}$ that satisfy the equation?

A: Yes, there may be other values of ${$ x$}$ and ${$ y$}$ that satisfy the equation, in addition to ${$ x=1$}$ and ${$ y=4$}$.

References

[1] Algebra, 2nd ed. by Michael Artin
[2] Calculus, 3rd ed. by Michael Spivak
Understanding Points on the Graph of an Equation: A Q&A Guide ================================================================

In our previous article, we explored the concept of points on the graph of an equation and determined which statement must be true if the point ${$ (1,4)$}$ is on the graph. In this article, we'll continue to delve deeper into the topic and answer some frequently asked questions.

Q: What is the relationship between a point on the graph and the equation?

A: A point ${$ (x,y)$}$ is said to be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ satisfy the equation. In other words, when we substitute the values of ${$ x$}$ and ${$ y$}$ into the equation, the equation holds true.

Q: Can a point be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ do not satisfy the equation?

A: No, a point cannot be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ do not satisfy the equation. By definition, a point on the graph of an equation must satisfy the equation.

Q: What is the difference between a point on the graph and a point that is not on the graph?

A: A point on the graph of an equation satisfies the equation, while a point that is not on the graph does not satisfy the equation. In other words, a point on the graph is a solution to the equation, while a point that is not on the graph is not a solution.

Q: Can a point be on the graph of an equation if the equation is not true?

A: No, a point cannot be on the graph of an equation if the equation is not true. By definition, a point on the graph of an equation must satisfy the equation, and if the equation is not true, then the point cannot be on the graph.

Q: What is the significance of a point being on the graph of an equation?

A: A point being on the graph of an equation is significant because it means that the values of ${$ x$}$ and ${$ y$}$ satisfy the equation. This is useful in a variety of applications, such as solving systems of equations, graphing functions, and modeling real-world phenomena.

Q: Can a point be on the graph of an equation if the equation is a linear equation?

A: Yes, a point can be on the graph of a linear equation. In fact, the graph of a linear equation is a line, and a point on the graph of a linear equation is a point on that line.

Q: Can a point be on the graph of an equation if the equation is a quadratic equation?

A: Yes, a point can be on the graph of a quadratic equation. In fact, the graph of a quadratic equation is a parabola, and a point on the graph of a quadratic equation is a point on that parabola.

Q: Can a point be on the graph of an equation if the equation is a polynomial equation?

A: Yes, a point can be on the graph of a polynomial equation. In fact, the graph of a polynomial equation is a curve, and a point on the graph of a polynomial equation is a point on that curve.

Q: Can a point be on the graph of an equation if the equation is a rational equation?

A: Yes, a point can be on the graph of a rational equation. In fact, the graph of a rational equation is a curve, and a point on the graph of a rational equation is a point on that curve.

Conclusion

In conclusion, a point on the graph of an equation is a point that satisfies the equation. This is a fundamental concept in mathematics, and it has a wide range of applications in fields such as algebra, geometry, and calculus. We hope that this Q&A guide has helped to clarify the concept of points on the graph of an equation and has provided a useful resource for students and educators.

Frequently Asked Questions

Q: What is the relationship between a point on the graph and the equation?

A: A point ${$ (x,y)$}$ is said to be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ satisfy the equation.

Q: Can a point be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ do not satisfy the equation?

A: No, a point cannot be on the graph of an equation if the values of ${$ x$}$ and ${$ y$}$ do not satisfy the equation.

Q: What is the significance of a point being on the graph of an equation?

A: A point being on the graph of an equation is significant because it means that the values of ${$ x$}$ and ${$ y$}$ satisfy the equation.

References

[1] Algebra, 2nd ed. by Michael Artin
[2] Calculus, 3rd ed. by Michael Spivak
[3] Geometry, 2nd ed. by Michael Spivak