Determine If The Point $(-1, 3)$ Lies On The Line Given By The Equation $-2x + Y = -4$. Answer: $\qquad$

Introduction

In mathematics, the concept of lines and points is a fundamental aspect of geometry. A line is a set of points that extend infinitely in two directions, and a point is a location in space that has no dimension. In this article, we will explore the problem of determining whether a given point lies on a line, using the equation of the line and the coordinates of the point.

The Equation of a Line

The equation of a line is typically written in the form:

$$ax + by = c$$

where $a$, $b$, and $c$ are constants, and $x$ and $y$ are the variables representing the coordinates of a point on the line. In this case, the equation of the line is:

$$-2x + y = -4$$

The Point in Question

The point in question is given as $(-1, 3)$. This means that the $x$-coordinate of the point is $-1$, and the $y$-coordinate is $3$.

Substituting the Point into the Equation

To determine whether the point lies on the line, we need to substitute the coordinates of the point into the equation of the line. This means replacing $x$ with $-1$ and $y$ with $3$ in the equation:

$$-2(-1) + 3 = -4$$

Simplifying the Equation

Now, let's simplify the equation by performing the arithmetic operations:

$$2 + 3 = -4$$

$$5 = -4$$

Conclusion

As we can see, the equation does not hold true, since $5$ is not equal to $-4$. This means that the point $(-1, 3)$ does not lie on the line given by the equation $-2x + y = -4$.

Why This Matters

Understanding whether a point lies on a line is an important concept in mathematics, with applications in various fields such as physics, engineering, and computer science. For example, in physics, the concept of lines and points is used to describe the motion of objects, while in engineering, it is used to design and analyze systems.

Real-World Applications

The concept of lines and points has numerous real-world applications, including:

• Computer-Aided Design (CAD): In CAD, lines and points are used to create and manipulate 2D and 3D models of objects.
• Computer Graphics: In computer graphics, lines and points are used to create and animate 2D and 3D graphics.
• Physics and Engineering: In physics and engineering, lines and points are used to describe the motion of objects and design and analyze systems.

Conclusion

In conclusion, determining whether a point lies on a line is a fundamental concept in mathematics, with numerous real-world applications. By understanding the equation of a line and the coordinates of a point, we can determine whether the point lies on the line. In this article, we used the equation $-2x + y = -4$ and the point $(-1, 3)$ to demonstrate this concept.

Additional Resources

For further reading and practice, we recommend the following resources:

• Mathematics textbooks: There are many excellent mathematics textbooks that cover the concept of lines and points, including "Calculus" by Michael Spivak and "Linear Algebra and Its Applications" by Gilbert Strang.
• Online resources: There are many online resources available that provide interactive lessons and exercises on the concept of lines and points, including Khan Academy and MIT OpenCourseWare.

Final Thoughts

Q: What is the equation of a line?

A: The equation of a line is typically written in the form:

$$ax + by = c$$

where $a$, $b$, and $c$ are constants, and $x$ and $y$ are the variables representing the coordinates of a point on the line.

Q: How do I determine if a point lies on a line?

A: To determine if a point lies on a line, you need to substitute the coordinates of the point into the equation of the line. If the equation holds true, then the point lies on the line.

Q: What if the equation does not hold true?

A: If the equation does not hold true, then the point does not lie on the line.

Q: Can a point lie on a line if the equation does not hold true?

A: No, a point cannot lie on a line if the equation does not hold true.

Q: What if the point has the same x-coordinate as the line?

A: If the point has the same x-coordinate as the line, then you need to substitute the x-coordinate into the equation of the line and solve for y.

Q: What if the point has the same y-coordinate as the line?

A: If the point has the same y-coordinate as the line, then you need to substitute the y-coordinate into the equation of the line and solve for x.

Q: Can a point lie on a line if it has the same x-coordinate or y-coordinate as the line?

A: Yes, a point can lie on a line if it has the same x-coordinate or y-coordinate as the line.

Q: How do I find the equation of a line given two points?

A: To find the equation of a line given two points, you need to use the slope-intercept form of the equation of a line:

$$y = mx + b$$

where $m$ is the slope of the line and $b$ is the y-intercept.

Q: How do I find the slope of a line given two points?

A: To find the slope of a line given two points, you need to use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Q: How do I find the y-intercept of a line given two points?

A: To find the y-intercept of a line given two points, you need to use the equation:

$$b = y_1 - mx_1$$

where $(x_1, y_1)$ is the coordinate of one of the points and $m$ is the slope of the line.

Q: Can a line have a y-intercept of zero?

A: Yes, a line can have a y-intercept of zero.

Q: Can a line have a slope of zero?

A: Yes, a line can have a slope of zero.

Q: What is the difference between a line and a plane?

A: A line is a set of points that extend infinitely in two directions, while a plane is a set of points that extend infinitely in three directions.

Q: Can a point lie on a plane?

A: Yes, a point can lie on a plane.

Q: Can a line lie on a plane?

A: Yes, a line can lie on a plane.

Q: Can a plane have a line that lies on it?

A: Yes, a plane can have a line that lies on it.

Conclusion

In conclusion, determining if a point lies on a line is a fundamental concept in mathematics, with numerous real-world applications. By understanding the equation of a line and the coordinates of a point, we can determine whether the point lies on the line. We hope that this article has provided a clear and concise explanation of this concept, and we encourage readers to practice and explore this topic further.