The Point-slope Form Of The Equation Of The Line That Passes Through { (-5, -1)$}$ And { (10, -7)$}$ Is ${ Y + 7 = -\frac{2}{5}(x - 10). }$What Is The Standard Form Of The Equation For This Line?A. [$2x - 5y =

Introduction

The point-slope form of a line is a mathematical representation that uses the slope of the line and a point on the line to define its equation. This form is particularly useful when we know the slope and a point on the line, as it allows us to easily write the equation of the line. In this article, we will explore the point-slope form of a line and learn how to convert it to the standard form.

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. This form is called the point-slope form because it uses the slope and a point on the line to define the equation.

Example: Finding the Point-Slope Form

Let's consider an example to see how to use the point-slope form. Suppose we know that the line passes through the points $(-5, -1)$ and $(10, -7)$. We can use these points to find the slope of the line, which is given by:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-7 - (-1)}{10 - (-5)} = \frac{-6}{15} = -\frac{2}{5}$$

Now that we have the slope, we can use the point-slope form to write the equation of the line. Let's use the point $(-5, -1)$ as the point on the line. Then, the equation of the line is:

$$y - (-1) = -\frac{2}{5}(x - (-5))$$

Simplifying this equation, we get:

$$y + 1 = -\frac{2}{5}(x + 5)$$

This is the point-slope form of the equation of the line.

Converting to Standard Form

The standard form of a line is given by the equation:

$$Ax + By = C$$

where $A$, $B$, and $C$ are constants. To convert the point-slope form to the standard form, we need to eliminate the fractions and simplify the equation.

Example: Converting to Standard Form

Let's convert the point-slope form of the equation of the line to the standard form. We have:

$$y + 1 = -\frac{2}{5}(x + 5)$$

To eliminate the fractions, we can multiply both sides of the equation by 5:

$$5(y + 1) = -2(x + 5)$$

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

$$5y + 5 = -2x - 10$$

Now, let's simplify the equation by moving all the terms to the left-hand side:

$$5y + 2x + 15 = 0$$

This is the standard form of the equation of the line.

Conclusion

In this article, we learned how to use the point-slope form of a line to find the equation of a line that passes through two points. We also learned how to convert the point-slope form to the standard form. The point-slope form is a useful tool for finding the equation of a line, and it is particularly useful when we know the slope and a point on the line.

The Standard Form of the Equation

The standard form of the equation of the line is:

$$2x - 5y = 15$$

This is the standard form of the equation of the line that passes through the points $(-5, -1)$ and $(10, -7)$.

Discussion

The point-slope form of a line is a powerful tool for finding the equation of a line. It is particularly useful when we know the slope and a point on the line. The standard form of the equation of the line is also useful, as it allows us to easily identify the slope and the y-intercept of the line.

References

• [1] "Point-Slope Form of a Line" by Math Open Reference
• [2] "Standard Form of a Line" by Math Is Fun

Additional Resources

• [1] "Point-Slope Form of a Line" by Khan Academy
• [2] "Standard Form of a Line" by Purplemath
  The Point-Slope Form of a Line: Q&A =====================================

Introduction

In our previous article, we explored the point-slope form of a line and learned how to convert it to the standard form. In this article, we will answer some common questions about the point-slope form of a line.

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

A: The point-slope form of a line is a mathematical representation that uses the slope of the line and a point on the line to define its equation. It 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 need to know two points on the line. Let's say the two points are $(x_1, y_1)$ and $(x_2, y_2)$. The slope of the line is given by:

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

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 eliminate the fractions and simplify the equation. Here's a step-by-step guide:

1. Multiply both sides of the equation by the denominator of the fraction.
2. Expand the left-hand side of the equation.
3. Move all the terms to the left-hand side of the equation.
4. Simplify 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 slope and y-intercept of a line using the standard form?

A: To find the slope and y-intercept of a line using the standard form, you need to rewrite the equation in the form:

$$y = mx + b$$

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

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

A: Here are some common mistakes to avoid when working with the point-slope form:

• Not eliminating the fractions when converting to the standard form.
• Not simplifying the equation when converting to the standard form.
• Not using the correct formula for the slope.
• Not checking the signs of the coefficients when rewriting the equation in the standard form.

Conclusion

In this article, we answered some common questions about the point-slope form of a line. We hope that this article has been helpful in clarifying any confusion you may have had about the point-slope form. If you have any further questions, please don't hesitate to ask.

Additional Resources

• [1] "Point-Slope Form of a Line" by Khan Academy
• [2] "Standard Form of a Line" by Purplemath
• [3] "Point-Slope Form of a Line" by Math Is Fun

Discussion

The point-slope form of a line is a powerful tool for finding the equation of a line. It is particularly useful when we know the slope and a point on the line. The standard form of the equation of a line is also useful, as it allows us to easily identify the slope and the y-intercept of the line.

References

• [1] "Point-Slope Form of a Line" by Math Open Reference
• [2] "Standard Form of a Line" by Math Is Fun

Glossary

• Point-slope form: A mathematical representation that uses the slope of the line and a point on the line to define its equation.
• Standard form: A mathematical representation that uses the slope and the y-intercept to define the equation of a line.
• Slope: A measure of how steep a line is.
• Y-intercept: The point where the line intersects the y-axis.