Write The Equation Of The Line That Passes Through The Points { (8, -1)$}$ And { (2, -5)$}$ In Standard Form, Given That The Point-slope Form Is ${ Y + 1 = \frac{2}{3}(x - 8). } F I L L I N T H E B L A N K S : Fill In The Blanks: F I Ll In T H E B L Ank S : [ \square , X +

Introduction

In mathematics, the equation of a line can be expressed in various forms, including standard form, point-slope form, and slope-intercept form. The standard form of a line is given by the equation $Ax + By = C$, where $A$, $B$, and $C$ are constants. In this article, we will focus on finding the equation of a line that passes through two given points in standard form, given that the point-slope form is already provided.

Understanding the Point-Slope Form

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. In the given problem, the point-slope form is $y + 1 = \frac{2}{3}(x - 8)$. This equation represents the line that passes through the point $(8, -1)$ and has a slope of $\frac{2}{3}$.

Converting Point-Slope Form to Standard Form

To convert the point-slope form to standard form, we need to isolate the $y$ term and simplify the equation. We can start by multiplying both sides of the equation by $3$ to eliminate the fraction:

$$3(y + 1) = 2(x - 8)$$

Expanding the left-hand side of the equation, we get:

$$3y + 3 = 2x - 16$$

Now, we can rearrange the terms to get the equation in standard form:

$$2x - 3y = 19$$

Filling in the Blanks

The standard form of the equation of a line is given by $Ax + By = C$. Comparing this with the equation we obtained in the previous step, we can fill in the blanks:

$$\boxed{2} \, x - \boxed{3} \, y = \boxed{19}$$

Conclusion

In this article, we have shown how to convert the point-slope form of a line to standard form. We started with the given point-slope form $y + 1 = \frac{2}{3}(x - 8)$ and converted it to standard form $2x - 3y = 19$. We also filled in the blanks to obtain the final equation in standard form.

Key Takeaways

• The standard form of a line is given by $Ax + By = C$.
• The point-slope form of a line is given by $y - y_1 = m(x - x_1)$.
• To convert the point-slope form to standard form, we need to isolate the $y$ term and simplify the equation.
• We can fill in the blanks by comparing the equation with the standard form $Ax + By = C$.

Practice Problems

1. Find the equation of the line that passes through the points $(4, 2)$ and $(6, 3)$ in standard form, given that the point-slope form is $y - 2 = \frac{1}{2}(x - 4)$.
2. Find the equation of the line that passes through the points $(1, 4)$ and $(3, 6)$ in standard form, given that the point-slope form is $y - 4 = 2(x - 1)$.

Solutions

1. To find the equation of the line in standard form, we can follow the same steps as before. We start by multiplying both sides of the equation by $2$ to eliminate the fraction:

$$2(y - 2) = 1(x - 4)$$

Expanding the left-hand side of the equation, we get:

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

Now, we can rearrange the terms to get the equation in standard form:

$$x - 2y = 0$$

2. To find the equation of the line in standard form, we can follow the same steps as before. We start by multiplying both sides of the equation by $2$ to eliminate the fraction:

$$2(y - 4) = 2(x - 1)$$

Expanding the left-hand side of the equation, we get:

$$2y - 8 = 2x - 2$$

Now, we can rearrange the terms to get the equation in standard form:

2x - 2y = 6$<br/> **Frequently Asked Questions (FAQs) about Converting Point-Slope Form to Standard Form** =====================================================================================

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 convert the point-slope form to standard form?

A: To convert the point-slope form to standard form, you need to isolate the $y$ term and simplify the equation. You can start by multiplying both sides of the equation by a constant to eliminate the fraction, and then rearrange the terms to get the equation in standard form.

Q: What is the standard form of a line?

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

Q: How do I fill in the blanks in the standard form equation?

A: To fill in the blanks in the standard form equation, you need to compare the equation with the standard form $Ax + By = C$. You can identify the values of $A$, $B$, and $C$ by looking at the coefficients of $x$ and $y$ and the constant term.

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

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

• Not isolating the $y$ term
• Not simplifying the equation
• Not rearranging the terms to get the equation in standard form
• Not filling in the blanks correctly

Q: Can I use a calculator to convert point-slope form to standard form?

A: Yes, you can use a calculator to convert point-slope form to standard form. However, it's always a good idea to double-check your work by hand to make sure you understand the process.

Q: How do I check my work when converting point-slope form to standard form?

A: To check your work when converting point-slope form to standard form, you can plug in the values of $x$ and $y$ from the original equation into the standard form equation and make sure they are equal.

Q: What are some real-world applications of converting point-slope form to standard form?

A: Some real-world applications of converting point-slope form to standard form include:

• Finding the equation of a line that passes through two given points
• Graphing a line on a coordinate plane
• Solving systems of linear equations
• Finding the slope and y-intercept of a line

Q: Can I use the same process to convert slope-intercept form to standard form?

A: Yes, you can use the same process to convert slope-intercept form to standard form. However, you will need to isolate the $y$ term and simplify the equation before rearranging the terms to get the equation in standard form.

Q: What are some tips for mastering the process of converting point-slope form to standard form?

A: Some tips for mastering the process of converting point-slope form to standard form include:

• Practicing, practicing, practicing!
• Paying close attention to the coefficients of $x$ and $y$ and the constant term
• Using a calculator to check your work
• Double-checking your work by hand to make sure you understand the process

Conclusion

Converting point-slope form to standard form is an important skill in algebra that can be used to solve a variety of problems. By following the steps outlined in this article and practicing regularly, you can master the process and become more confident in your ability to solve equations and graph lines.