Applying the concept to the given equation

Oct 13, 2025 by magicalceylontours.com 60 views

Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables, x and y. The equation of a line can be written in various forms, including the slope-intercept form, the standard form, and the general form. In this article, we will focus on finding the y-intercept of a line whose equation is given in the general form.

What is the y-intercept?

The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero. In other words, it is the point on the line where the x-coordinate is zero. The y-intercept is an important concept in mathematics, as it provides a way to determine the value of y when x is zero.

Finding the y-intercept

To find the y-intercept of a line whose equation is given in the general form, we need to isolate the variable y. The general form of a line is given by:

ax + by = c

where a, b, and c are constants. To find the y-intercept, we need to isolate y by moving all the terms involving x to the other side of the equation.

Step 1: Move all the terms involving x to the other side of the equation

Let's start by moving all the terms involving x to the other side of the equation. We can do this by subtracting ax from both sides of the equation:

by = -ax + c

Step 2: Divide both sides of the equation by b

Next, we need to divide both sides of the equation by b to isolate y:

y = (-a/b)x + c/b

Step 3: Find the value of y when x is equal to zero

Now that we have isolated y, we can find the value of y when x is equal to zero. We can do this by substituting x = 0 into the equation:

y = (-a/b)(0) + c/b

y = c/b

The y-intercept of the line

The y-intercept of the line is the value of y when x is equal to zero. In this case, the y-intercept is c/b.

Applying the concept to the given equation

Now that we have found the y-intercept, let's apply the concept to the given equation:

7x - 4y = 8

We can rewrite the equation in the general form as:

7x - 4y - 8 = 0

Now, we can find the y-intercept by following the steps we outlined earlier:

Step 1: Move all the terms involving x to the other side of the equation:

-4y = -7x + 8

Step 2: Divide both sides of the equation by -4:

y = (7/4)x - 2

Step 3: Find the value of y when x is equal to zero:

y = (7/4)(0) - 2

y = -2

Conclusion

In this article, we have discussed the concept of the y-intercept of a line and how to find it when the equation of the line is given in the general form. We have applied the concept to the given equation and found the y-intercept to be -2. The y-intercept is an important concept in mathematics, as it provides a way to determine the value of y when x is zero.

References

[1] Khan Academy. (n.d.). Slope-Intercept Form. Retrieved from https://www.khanacademy.org/math/algebra/x2f0f7f/slope-intercept-form
[2] Math Open Reference. (n.d.). General Form of a Line. Retrieved from https://www.mathopenref.com/linegeneralform.html

Discussion

What is the y-intercept of the line whose equation is 7x - 4y = 8? Do you have any questions about the concept of the y-intercept? Share your thoughts and questions in the comments below!

Answer Key

A. 8 B. -2 C. 7/4

Correct Answer

Introduction

In our previous article, we discussed the concept of the y-intercept of a line and how to find it when the equation of the line is given in the general form. We applied the concept to the given equation 7x - 4y = 8 and found the y-intercept to be -2. In this article, we will answer some frequently asked questions about the y-intercept.

Q&A

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero.

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

A: To find the y-intercept of a line, you need to isolate the variable y by moving all the terms involving x to the other side of the equation. Then, you can find the value of y when x is equal to zero.

Q: What is the general form of a line?

A: The general form of a line is given by ax + by = c, where a, b, and c are constants.

Q: How do I rewrite the equation of a line in the general form?

A: To rewrite the equation of a line in the general form, you need to move all the terms involving x to the other side of the equation. Then, you can rewrite the equation in the form ax + by = c.

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

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

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

A: To find the slope of a line, you need to rewrite the equation of the line in the slope-intercept form. Then, you can find the slope by looking at the coefficient of x.

Q: What is the relationship between the y-intercept and the slope of a line?

A: The y-intercept and the slope of a line are related in that the y-intercept is the value of y when x is equal to zero, and the slope is the rate of change of y with respect to x.

Q: Can you give an example of how to find the y-intercept of a line?

A: Yes, let's consider the equation 2x + 3y = 5. To find the y-intercept, we need to isolate the variable y by moving all the terms involving x to the other side of the equation:

3y = -2x + 5

Then, we can find the value of y when x is equal to zero:

y = (-2/3)(0) + 5/3

y = 5/3

Q: Can you give an example of how to find the slope of a line?

A: Yes, let's consider the equation y = 2x + 3. To find the slope, we can look at the coefficient of x, which is 2. Therefore, the slope of the line is 2.

Q: What is the significance of the y-intercept in real-world applications?

A: The y-intercept is an important concept in real-world applications, as it provides a way to determine the value of y when x is zero. For example, in economics, the y-intercept can represent the initial cost of a product, while the slope can represent the rate of change of the cost with respect to the quantity produced.

Conclusion

In this article, we have answered some frequently asked questions about the y-intercept of a line. We have discussed the concept of the y-intercept, how to find it, and its relationship with the slope of a line. We have also provided examples of how to find the y-intercept and the slope of a line. We hope that this article has been helpful in clarifying any doubts you may have had about the y-intercept.

References

[1] Khan Academy. (n.d.). Slope-Intercept Form. Retrieved from https://www.khanacademy.org/math/algebra/x2f0f7f/slope-intercept-form
[2] Math Open Reference. (n.d.). General Form of a Line. Retrieved from https://www.mathopenref.com/linegeneralform.html

Discussion

Do you have any questions about the y-intercept or the slope of a line? Share your thoughts and questions in the comments below!

Answer Key

Q: What is the y-intercept of a line? A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero.
Q: How do I find the y-intercept of a line? A: To find the y-intercept of a line, you need to isolate the variable y by moving all the terms involving x to the other side of the equation. Then, you can find the value of y when x is equal to zero.
Q: What is the general form of a line? A: The general form of a line is given by ax + by = c, where a, b, and c are constants.
Q: How do I rewrite the equation of a line in the general form? A: To rewrite the equation of a line in the general form, you need to move all the terms involving x to the other side of the equation. Then, you can rewrite the equation in the form ax + by = c.
Q: What is the slope-intercept form of a line? A: The slope-intercept form of a line is given by y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the slope of a line? A: To find the slope of a line, you need to rewrite the equation of the line in the slope-intercept form. Then, you can find the slope by looking at the coefficient of x.
Q: What is the relationship between the y-intercept and the slope of a line? A: The y-intercept and the slope of a line are related in that the y-intercept is the value of y when x is equal to zero, and the slope is the rate of change of y with respect to x.
Q: Can you give an example of how to find the y-intercept of a line? A: Yes, let's consider the equation 2x + 3y = 5. To find the y-intercept, we need to isolate the variable y by moving all the terms involving x to the other side of the equation: 3y = -2x + 5 Then, we can find the value of y when x is equal to zero: y = (-2/3)(0) + 5/3 y = 5/3
Q: Can you give an example of how to find the slope of a line? A: Yes, let's consider the equation y = 2x + 3. To find the slope, we can look at the coefficient of x, which is 2. Therefore, the slope of the line is 2.
Q: What is the significance of the y-intercept in real-world applications? A: The y-intercept is an important concept in real-world applications, as it provides a way to determine the value of y when x is zero. For example, in economics, the y-intercept can represent the initial cost of a product, while the slope can represent the rate of change of the cost with respect to the quantity produced.