The Point-slope Form Of The Equation Of The Line That Passes Through $(-4, -3)$ And $(12, 1)$ Is $y - 1 = \frac{1}{4}(x - 12)$. What Is The Standard Form Of The Equation For This Line?A. $x - 4y =

Introduction

In mathematics, the point-slope form of a line is a powerful tool used to describe the equation of a line that passes through two given points. The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. In this article, we will explore the conversion of the point-slope form to the standard form of the equation of a line.

Understanding the Point-Slope Form

The point-slope form of the equation of a line is given by the equation $y - y_1 = m(x - x_1)$. This form is useful when we know the coordinates of a point on the line and the slope of the line. The slope of the line, $m$, can be calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Converting Point-Slope Form to Standard Form

To convert the point-slope form to the standard form, we need to simplify the equation by isolating the $x$ and $y$ terms. The standard form of the equation of a line is given by the equation $Ax + By = C$, where $A$, $B$, and $C$ are constants.

Let's consider the point-slope form of the equation of the line that passes through $(-4, -3)$ and $(12, 1)$, which is given by the equation $y - 1 = \frac{1}{4}(x - 12)$. To convert this equation to the standard form, we need to simplify it by isolating the $x$ and $y$ terms.

Step 1: Simplify the Equation

The first step is to simplify the equation by distributing the slope, $\frac{1}{4}$, to the terms inside the parentheses.

$$y - 1 = \frac{1}{4}(x - 12)$$

$$y - 1 = \frac{1}{4}x - 3$$

Step 2: Isolate the $y$ Term

The next step is to isolate the $y$ term by adding $1$ to both sides of the equation.

$$y - 1 + 1 = \frac{1}{4}x - 3 + 1$$

$$y = \frac{1}{4}x - 2$$

Step 3: Multiply Both Sides by 4

To eliminate the fraction, we need to multiply both sides of the equation by $4$.

$$4y = 4(\frac{1}{4}x - 2)$$

$$4y = x - 8$$

Step 4: Rearrange the Equation

The final step is to rearrange the equation to the standard form by moving the $x$ term to the left-hand side of the equation.

$$x - 4y = 8$$

Conclusion

In this article, we have explored the conversion of the point-slope form to the standard form of the equation of a line. We have used the point-slope form of the equation of the line that passes through $(-4, -3)$ and $(12, 1)$ as an example. By simplifying the equation and isolating the $x$ and $y$ terms, we have arrived at the standard form of the equation, which is $x - 4y = 8$.

Discussion

The point-slope form and the standard form are two important forms of the equation of a line. The point-slope form is useful when we know the coordinates of a point on the line and the slope of the line, while the standard form is useful when we need to find the equation of a line in a specific format. By understanding the conversion between these two forms, we can solve a wide range of problems involving the equation of a line.

Example Problems

1. Find the standard form of the equation of the line that passes through $(2, 3)$ and $(4, 5)$.
2. Find the standard form of the equation of the line that passes through $(1, 2)$ and $(3, 4)$.
3. Find the standard form of the equation of the line that passes through $(0, 0)$ and $(2, 3)$.

Answer Key

1. $x - 2y = 1$
2. $x - 2y = 1$
3. $x - 3y = 6$

References

1. Algebra and Trigonometry by Michael Sullivan
2. College Algebra by James Stewart
3. Mathematics for Calculus by Michael Sullivan
   Frequently Asked Questions: Point-Slope Form to Standard Form Conversion ====================================================================

Q: What is the point-slope form of a line?

A: The point-slope form of a line is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line.

Q: How do I find the slope of a line using the point-slope form?

A: To find the slope of a line using the point-slope form, you can use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: How do I convert the point-slope form to the standard form?

A: To convert the point-slope form to the standard form, you need to simplify the equation by isolating the $x$ and $y$ terms. This involves distributing the slope, isolating the $y$ term, multiplying both sides by a constant, and rearranging the equation.

Q: What is the standard form of the equation of a line?

A: The standard form of the equation of a line is given by the equation $Ax + By = C$, where $A$, $B$, and $C$ are constants.

Q: How do I find the equation of a line in standard form?

A: To find the equation of a line in standard form, you can use the point-slope form and convert it to the standard form by following the steps outlined above.

Q: What are some common mistakes to avoid when converting the point-slope form to the standard form?

A: Some common mistakes to avoid when converting the point-slope form to the standard form include:

• Not distributing the slope correctly
• Not isolating the $y$ term correctly
• Not multiplying both sides by a constant correctly
• Not rearranging the equation correctly

Q: How do I check if my answer is correct?

A: To check if your answer is correct, you can plug in the values of $x$ and $y$ into the equation and see if it satisfies the equation.

Q: What are some real-world applications of the point-slope form and standard form?

A: The point-slope form and standard form have many real-world applications, including:

• Finding the equation of a line that passes through two points
• Finding the equation of a line that is perpendicular to another line
• Finding the equation of a line that is parallel to another line
• Solving systems of linear equations

Q: How do I use the point-slope form and standard form to solve systems of linear equations?

A: To use the point-slope form and standard form to solve systems of linear equations, you can:

• Use the point-slope form to find the equation of one line
• Use the standard form to find the equation of another line
• Solve the system of linear equations by finding the intersection of the two lines

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

• Not using the correct method to solve the system
• Not checking if the solution satisfies both equations
• Not considering the possibility of no solution or infinitely many solutions

Conclusion

In this article, we have answered some frequently asked questions about the point-slope form and standard form of the equation of a line. We have also discussed some common mistakes to avoid when converting the point-slope form to the standard form and solving systems of linear equations. By following the steps outlined above and avoiding common mistakes, you can master the point-slope form and standard form and use them to solve a wide range of problems involving the equation of a line.