Moving Terms to the Other Side

Oct 13, 2025 by pynkocean.com 65 views

Which Equation Represents a Line That Passes Through a Given Point and Has a Specific Slope?

Understanding the Problem

When it comes to graphing lines, there are several equations that can be used to represent a line. However, when we are given a specific point that the line passes through and a specific slope, we need to use the point-slope form of a linear equation to find the correct equation. In this article, we will explore how to use the point-slope form to find the equation of a line that passes through a given point and has a specific slope.

The Point-Slope Form of a Linear Equation

The point-slope form of a linear equation is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line. This form is useful when we are given a specific point that the line passes through and a specific slope.

Given Information

In this problem, we are given a point (4, 1/3) and a slope of 3/4. We need to use the point-slope form to find the equation of the line that passes through this point and has this slope.

Using the Point-Slope Form

To use the point-slope form, we need to substitute the given point and slope into the equation. We will substitute (x1, y1) = (4, 1/3) and m = 3/4 into the equation.

y - 1/3 = 3/4(x - 4)

Simplifying the Equation

To simplify the equation, we can start by multiplying both sides of the equation by 12 to eliminate the fractions.

12(y - 1/3) = 12(3/4)(x - 4)

This simplifies to:

12y - 4 = 9(x - 4)

Expanding and Simplifying

Next, we can expand and simplify the equation by distributing the 9 to the terms inside the parentheses.

12y - 4 = 9x - 36

Moving Terms to the Other Side

To isolate y, we can move the terms involving x to the other side of the equation.

12y = 9x - 36 + 4

This simplifies to:

12y = 9x - 32

Dividing Both Sides

Finally, we can divide both sides of the equation by 12 to solve for y.

y = (9/12)x - 32/12

This simplifies to:

y = (3/4)x - 8/3

Conclusion

In this article, we used the point-slope form of a linear equation to find the equation of a line that passes through a given point and has a specific slope. We substituted the given point and slope into the equation and simplified the result to find the final equation. The final equation is:

y = (3/4)x - 8/3

This equation represents a line that passes through the point (4, 1/3) and has a slope of 3/4.

Answer

The correct answer is:

C. y - 1/3 = 3/4(x - 4)

However, we simplified the equation to:

y = (3/4)x - 8/3

This is the final equation that represents a line that passes through the point (4, 1/3) and has a slope of 3/4.
Frequently Asked Questions (FAQs) About the Point-Slope Form of a Linear Equation

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 - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form, you need to substitute the given point and slope into the equation. You will substitute (x1, y1) = (4, 1/3) and m = 3/4 into the equation.

Q: What if I have a point and a slope, but I don't know the equation of the line?

A: In that case, you can use the point-slope form to find the equation of the line. Simply substitute the given point and slope into the equation and simplify the result.

Q: Can I use the point-slope form to find the equation of a line that passes through two points?

A: Yes, you can use the point-slope form to find the equation of a line that passes through two points. However, you will need to find the slope of the line first using the formula:

m = (y2 - y1) / (x2 - x1)

Q: What if I have a line with a slope of 0?

A: If you have a line with a slope of 0, it means that the line is horizontal. In this case, the equation of the line will be of the form:

y = c

where c is a constant.

Q: What if I have a line with a slope of infinity?

A: If you have a line with a slope of infinity, it means that the line is vertical. In this case, the equation of the line will be of the form:

x = c

where c is a constant.

Q: Can I use the point-slope form to find the equation of a line that passes through a point and has a slope that is a fraction?

A: Yes, you can use the point-slope form to find the equation of a line that passes through a point and has a slope that is a fraction. Simply substitute the given point and slope into the equation and simplify the result.

Q: What if I have a line with a slope that is a negative fraction?

A: If you have a line with a slope that is a negative fraction, it means that the line is sloping downward. In this case, the equation of the line will be of the form:

y - y1 = -m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Q: Can I use the point-slope form to find the equation of a line that passes through a point and has a slope that is a decimal?

A: Yes, you can use the point-slope form to find the equation of a line that passes through a point and has a slope that is a decimal. Simply substitute the given point and slope into the equation and simplify the result.

Q: What if I have a line with a slope that is a negative decimal?

A: If you have a line with a slope that is a negative decimal, it means that the line is sloping downward. In this case, the equation of the line will be of the form:

y - y1 = -m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Conclusion

In this article, we have answered some of the most frequently asked questions about the point-slope form of a linear equation. We have covered topics such as how to use the point-slope form to find the equation of a line, how to find the equation of a line that passes through two points, and how to handle lines with slopes that are fractions or decimals. We hope that this article has been helpful in answering your questions about the point-slope form of a linear equation.