Write An Equation In Point-slope Form That Has The Same Slope As $y=\frac{5}{7}x+4$ And Passes Through The Point $(2,5)$.A. $y - 2 = 4(x - 5$\]B. $y - 5 = \frac{5}{7}(x - 2$\]C. $y - 2 = \frac{5}{7}(x -

Mar 1, 2025 by ADMIN 207 views

**Solving Equations in Point-Slope Form**

Introduction

In mathematics, the point-slope form of a linear equation is a powerful tool for solving equations and graphing lines. 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 how to write an equation in point-slope form that has the same slope as a given equation and passes through a given point.

Understanding the Problem

The problem asks us to write an equation in point-slope form that has the same slope as the equation $y=\frac{5}{7}x+4$ and passes through the point $(2,5)$ . To solve this problem, we need to understand the concept of slope and how to use it to write an equation in point-slope form.

Slope and Point-Slope Form

The slope of a line is a measure of how steep the line is. It is calculated by dividing the change in $y$ by the change in $x$ . In the equation $y=\frac{5}{7}x+4$ , the slope is $\frac{5}{7}$ . This means that for every one unit increase in $x$ , the value of $y$ increases by $\frac{5}{7}$ units.

To write an equation in point-slope form, we need to use the slope and a point on the line. 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.

Step 1: Identify the Slope

The slope of the given equation is $\frac{5}{7}$ . This is the value of $m$ that we will use in the point-slope form.

Step 2: Identify the Point

The point that the line passes through is $(2,5)$ . This is the value of $(x_1, y_1)$ that we will use in the point-slope form.

Step 3: Write the Equation in Point-Slope Form

Now that we have identified the slope and the point, we can write the equation in point-slope form. The equation is given by:

y - 5 = \frac{5}{7}(x - 2)

This equation has the same slope as the given equation and passes through the given point.

Conclusion

In this article, we have learned how to write an equation in point-slope form that has the same slope as a given equation and passes through a given point. We have used the concept of slope and the point-slope form to solve the problem. The equation we have written is:

y - 5 = \frac{5}{7}(x - 2)

This equation has the same slope as the given equation and passes through the given point.

Answer

The correct answer is:

B. $y - 5 = \frac{5}{7}(x - 2)$

Discussion

This problem requires the student to understand the concept of slope and how to use it to write an equation in point-slope form. The student must also be able to identify the slope and a point on the line and use this information to write the equation in point-slope form.

Tips and Tricks

Make sure to identify the slope and a point on the line before writing the equation in point-slope form.
Use the point-slope form to write the equation, rather than trying to write the equation in slope-intercept form first.
Check your work by plugging in the point and making sure that the equation is true.

Practice Problems

Write an equation in point-slope form that has the same slope as the equation $y=2x-3$ and passes through the point $(1,4)$ .
Write an equation in point-slope form that has the same slope as the equation $y=-\frac{3}{4}x+2$ and passes through the point $(3,0)$ .
Write an equation in point-slope form that has the same slope as the equation $y=\frac{2}{3}x-1$ and passes through the point $(2,3)$ .

Solutions

$y - 4 = 2(x - 1)$
$y - 0 = -\frac{3}{4}(x - 3)$
$y - 3 = \frac{2}{3}(x - 2)$
Frequently Asked Questions (FAQs) =====================================

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

A: The point-slope form of a linear equation 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?

A: To find the slope of a line, 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 write an equation in point-slope form?

A: To write an equation in point-slope form, you need to identify the slope and a point on the line. Then, you can use the point-slope form equation $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point and $m$ is the slope.

Q: What is the difference between the point-slope form and the slope-intercept form?

A: The point-slope form and the slope-intercept form are two different ways of writing a linear equation. The point-slope form is given by the equation $y - y_1 = m(x - x_1)$ , while the slope-intercept form is given by the equation $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: How do I convert an equation from point-slope form to slope-intercept form?

A: To convert an equation from point-slope form to slope-intercept form, you can add $y_1$ to both sides of the equation and then simplify. For example, if you have the equation $y - 2 = 2(x - 1)$ , you can add 2 to both sides to get $y = 2x$ .

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

A: To find the y-intercept of a line, you can set $x = 0$ in the equation and solve for $y$ . For example, if you have the equation $y = 2x + 3$ , you can set $x = 0$ to get $y = 3$ .

Q: What is the significance of the y-intercept?

A: The y-intercept is the point where the line intersects the y-axis. It is an important point on the line because it gives you the value of $y$ when $x = 0$ .

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

A: To graph a line using the point-slope form, you can use the point and the slope to find two points on the line. Then, you can draw a line through the two points to graph the line.

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

A: Some common mistakes to avoid when working with the point-slope form include:

Forgetting to identify the slope and a point on the line
Not using the correct equation for the point-slope form
Not simplifying the equation correctly
Not checking the work by plugging in the point and making sure that the equation is true.

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

A: The point-slope form has many real-world applications, including:

Finding the equation of a line that passes through two points
Graphing a line using the point-slope form
Finding the slope and y-intercept of a line
Solving systems of linear equations.

Conclusion

In this article, we have answered some frequently asked questions about the point-slope form of a linear equation. We have covered topics such as finding the slope, writing an equation in point-slope form, converting an equation from point-slope form to slope-intercept form, and graphing a line using the point-slope form. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the point-slope form.