Find The Slope Of The Line − 7 X = 9 − 14 Y -7x = 9 - 14y − 7 X = 9 − 14 Y .

Mar 13, 2025 by ADMIN 77 views

**Finding the Slope of a Line: A Step-by-Step Guide**

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Introduction

In mathematics, the slope of a line is a fundamental concept that helps us understand the relationship between two variables. It is a measure of how steep the line is and can be calculated using the coordinates of two points on the line. In this article, we will focus on finding the slope of a line given in the form of an equation. We will use the equation $-7x = 9 - 14y$ as an example and walk through the steps to find the slope.

Understanding the Equation

The given equation is $-7x = 9 - 14y$ . To find the slope, we need to rewrite the equation in the slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Rearranging the Equation

To rewrite the equation in slope-intercept form, we need to isolate $y$ on one side of the equation. We can start by adding $14y$ to both sides of the equation:

-7x + 14y = 9

Next, we can add $7x$ to both sides of the equation to get:

14y = 7x + 9

Now, we can divide both sides of the equation by $14$ to isolate $y$ :

y = \frac{7}{14}x + \frac{9}{14}

Simplifying the Equation

We can simplify the equation by dividing the numerator and denominator of the fraction by their greatest common divisor, which is $7$ . This gives us:

y = \frac{1}{2}x + \frac{9}{14}

Finding the Slope

Now that we have the equation in slope-intercept form, we can easily identify the slope, which is the coefficient of $x$ . In this case, the slope is $\frac{1}{2}$ .

Interpreting the Slope

The slope of a line tells us how steep the line is. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. In this case, the slope is positive, indicating that the line rises from left to right.

Conclusion

Finding the slope of a line is an essential skill in mathematics, and it can be applied to a wide range of problems. By following the steps outlined in this article, you can easily find the slope of a line given in the form of an equation. Remember to rewrite the equation in slope-intercept form and identify the slope as the coefficient of $x$ .

Example Problems

Problem 1

Find the slope of the line $2x + 3y = 5$ .

Solution

To find the slope, we need to rewrite the equation in slope-intercept form. We can start by adding $-2x$ to both sides of the equation:

3y = -2x + 5

Next, we can divide both sides of the equation by $3$ to isolate $y$ :

y = -\frac{2}{3}x + \frac{5}{3}

The slope of the line is $-\frac{2}{3}$ .

Problem 2

Find the slope of the line $x - 2y = 3$ .

Solution

To find the slope, we need to rewrite the equation in slope-intercept form. We can start by adding $2y$ to both sides of the equation:

x = 2y + 3

Next, we can subtract $3$ from both sides of the equation to get:

x - 3 = 2y

Now, we can divide both sides of the equation by $2$ to isolate $y$ :

y = \frac{1}{2}x - \frac{3}{2}

The slope of the line is $\frac{1}{2}$ .

Tips and Tricks

To find the slope of a line, you need to rewrite the equation in slope-intercept form.
The slope is the coefficient of $x$ in the slope-intercept form.
A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right.

Common Mistakes

Not rewriting the equation in slope-intercept form.
Not identifying the slope as the coefficient of $x$ .
Not considering the sign of the slope.

Real-World Applications

Finding the slope of a line has many real-world applications, including:

Physics: The slope of a line can be used to describe the motion of an object.
Engineering: The slope of a line can be used to design and build structures such as bridges and buildings.
Economics: The slope of a line can be used to analyze the relationship between two variables such as supply and demand.

Conclusion

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Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It can be calculated using the coordinates of two points on the line.

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

A: To find the slope of a line, you need to rewrite the equation in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

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

A: The slope-intercept form of a line is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: How do I identify the slope in the slope-intercept form?

A: To identify the slope in the slope-intercept form, you need to look at the coefficient of $x$ . The slope is the coefficient of $x$ .

Q: What is the significance of the slope?

A: The slope of a line tells us how steep the line is. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right.

Q: Can I find the slope of a line without rewriting the equation in slope-intercept form?

A: No, you cannot find the slope of a line without rewriting the equation in slope-intercept form.

Q: What are some common mistakes to avoid when finding the slope of a line?

A: Some common mistakes to avoid when finding the slope of a line include:

Not rewriting the equation in slope-intercept form.
Not identifying the slope as the coefficient of $x$ .
Not considering the sign of the slope.

Q: What are some real-world applications of finding the slope of a line?

A: Some real-world applications of finding the slope of a line include:

Physics: The slope of a line can be used to describe the motion of an object.
Engineering: The slope of a line can be used to design and build structures such as bridges and buildings.
Economics: The slope of a line can be used to analyze the relationship between two variables such as supply and demand.

Q: Can I use a calculator to find the slope of a line?

A: Yes, you can use a calculator to find the slope of a line. However, it is recommended to use a calculator only as a check and to verify your answer.

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

A: The slope is the coefficient of $x$ in the slope-intercept form, while the y-intercept is the constant term.

Q: Can I find the slope of a line if the equation is not in slope-intercept form?

A: Yes, you can find the slope of a line if the equation is not in slope-intercept form. You need to rewrite the equation in slope-intercept form first.

Q: What are some tips for finding the slope of a line?

A: Some tips for finding the slope of a line include:

Make sure to rewrite the equation in slope-intercept form.
Identify the slope as the coefficient of $x$ .
Consider the sign of the slope.
Use a calculator only as a check and to verify your answer.

Q: Can I find the slope of a line if the equation is a quadratic equation?

A: Yes, you can find the slope of a line if the equation is a quadratic equation. However, you need to rewrite the equation in slope-intercept form first.

Q: What are some common mistakes to avoid when finding the slope of a line in a quadratic equation?

A: Some common mistakes to avoid when finding the slope of a line in a quadratic equation include:

Not rewriting the equation in slope-intercept form.
Not identifying the slope as the coefficient of $x$ .
Not considering the sign of the slope.

Q: Can I find the slope of a line if the equation is a system of linear equations?

A: Yes, you can find the slope of a line if the equation is a system of linear equations. However, you need to solve the system of linear equations first and then rewrite the equation in slope-intercept form.

Q: What are some common mistakes to avoid when finding the slope of a line in a system of linear equations?

A: Some common mistakes to avoid when finding the slope of a line in a system of linear equations include:

Not solving the system of linear equations first.
Not rewriting the equation in slope-intercept form.
Not identifying the slope as the coefficient of $x$ .
Not considering the sign of the slope.