Substituting the Values into the Equation

Oct 13, 2025 by miodeya.com 50 views

Introduction

In mathematics, the slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept is the point at which the line intersects the y-axis, and it is denoted by the letter b. In this article, we will discuss how to find the y-intercept of a line passing through a given point with a known slope.

The Slope-Intercept Form of a Line

The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep the line is, and it is calculated as the ratio of the vertical change (the rise) to the horizontal change (the run). The y-intercept is the point at which the line intersects the y-axis, and it is denoted by the letter b.

Finding the y-Intercept

To find the y-intercept of a line passing through a given point with a known slope, we can use the slope-intercept form of the equation of a line. We are given the point (4, -5) and the slope -1/9. We can substitute these values into the equation y = mx + b and solve for b.

Substituting the Values into the Equation

We are given the point (4, -5) and the slope -1/9. We can substitute these values into the equation y = mx + b as follows:

y = (-1/9)x + b

We are also given that the point (4, -5) lies on the line. We can substitute the values of x and y into the equation as follows:

-5 = (-1/9)(4) + b

Solving for b

To solve for b, we can start by multiplying both sides of the equation by 9 to eliminate the fraction:

-45 = -4 + 9b

Next, we can add 4 to both sides of the equation to isolate the term with b:

-41 = 9b

Finally, we can divide both sides of the equation by 9 to solve for b:

b = -41/9

Conclusion

In this article, we discussed how to find the y-intercept of a line passing through a given point with a known slope. We used the slope-intercept form of the equation of a line and substituted the given values into the equation to solve for b. We found that the y-intercept of the line passing through the point (4, -5) with a slope of -1/9 is -41/9.

Answer

The correct answer is C. ${$ \frac{41}{9}$}$.

Discussion

The y-intercept of a line is an important concept in mathematics, and it is used to determine the point at which the line intersects the y-axis. In this article, we discussed how to find the y-intercept of a line passing through a given point with a known slope. We used the slope-intercept form of the equation of a line and substituted the given values into the equation to solve for b. We found that the y-intercept of the line passing through the point (4, -5) with a slope of -1/9 is -41/9.

References

[1] "Slope-Intercept Form of a Line" by Math Open Reference
[2] "Finding the Equation of a Line" by Mathway
[3] "Graphing a Line" by Khan Academy
[4] "Slope of a Line" by Purplemath
[5] "Y-Intercept of a Line" by IXL

Introduction

In our previous article, we discussed how to find the y-intercept of a line passing through a given point with a known slope. In this article, we will answer some frequently asked questions (FAQs) about the y-intercept of a line.

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 denoted by the letter b in the slope-intercept form of the equation of a line, y = mx + b.

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

A: To find the y-intercept of a line, you can use the slope-intercept form of the equation of a line, y = mx + b. Substitute the given values into the equation and solve for b.

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

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

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

A: To calculate the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

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

A: The slope of a line is a measure of how steep the line is, while the y-intercept is the point at which the line intersects the y-axis.

Q: Can I find the y-intercept of a line if I know the slope and two points on the line?

A: Yes, you can find the y-intercept of a line if you know the slope and two points on the line. Use the slope-intercept form of the equation of a line and substitute the given values into the equation to solve for b.

Q: How do I graph a line if I know the y-intercept and the slope?

A: To graph a line if you know the y-intercept and the slope, use the slope-intercept form of the equation of a line, y = mx + b. Plot the y-intercept on the y-axis and then use the slope to determine the direction of the line.

Q: Can I find the y-intercept of a line if I know the equation of the line in the standard form?

A: Yes, you can find the y-intercept of a line if you know the equation of the line in the standard form, Ax + By = C. Rearrange the equation to the slope-intercept form, y = mx + b, and then solve for b.

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

A: The y-intercept of a line is an important concept in mathematics, and it is used to determine the point at which the line intersects the y-axis. It is also used in various applications, such as physics, engineering, and economics.

Q: Can I find the y-intercept of a line if I know the equation of the line in the point-slope form?

A: Yes, you can find the y-intercept of a line if you know the equation of the line in the point-slope form, y - y1 = m(x - x1). Rearrange the equation to the slope-intercept form, y = mx + b, and then solve for b.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the y-intercept of a line. We discussed how to find the y-intercept of a line, the significance of the y-intercept, and how to graph a line if you know the y-intercept and the slope.

References

[1] "Slope-Intercept Form of a Line" by Math Open Reference
[2] "Finding the Equation of a Line" by Mathway
[3] "Graphing a Line" by Khan Academy
[4] "Slope of a Line" by Purplemath
[5] "Y-Intercept of a Line" by IXL