Finding the Equation of a Straight Line with a Given Gradient

Oct 13, 2025 by magicalceylontours.com 141 views

Introduction

In mathematics, the equation of a straight line can be determined using various methods, including the point-slope form and the slope-intercept form. The point-slope form is particularly useful when we are given a point on the line and the gradient of the line. In this article, we will explore how to find the equation of a straight line through a given point with a specified gradient.

Understanding the Point-Slope Form

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

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, and m is the gradient of the line. This form is useful because it allows us to easily find the equation of a line when we know a point on the line and the gradient.

Finding the Equation of a Straight Line with a Given Gradient

To find the equation of a straight line through the point (2, 3) with a gradient of -2/3, we can use the point-slope form of the equation of a straight line.

Step 1: Identify the Point and the Gradient

The point on the line is given as (2, 3), and the gradient of the line is -2/3.

Step 2: Substitute the Values into the Point-Slope Form

Substituting the values of the point and the gradient into the point-slope form, we get:

y - 3 = -2/3(x - 2)

Step 3: Simplify the Equation

To simplify the equation, we can multiply both sides by 3 to eliminate the fraction:

3(y - 3) = -2(x - 2)

Expanding the left-hand side, we get:

3y - 9 = -2x + 4

Step 4: Rearrange the Equation

To put the equation in the standard form, we can rearrange the terms:

2x + 3y = 13

Conclusion

Example Problems

Problem 1

Find the equation of the straight line through the point (4, 5) with a gradient of 2/3.

Solution

Using the point-slope form, we get:

y - 5 = 2/3(x - 4)

Multiplying both sides by 3, we get:

3(y - 5) = 2(x - 4)

Expanding the left-hand side, we get:

3y - 15 = 2x - 8

Rearranging the terms, we get:

2x - 3y = -7

Problem 2

Find the equation of the straight line through the point (1, 2) with a gradient of -1.

Solution

Using the point-slope form, we get:

y - 2 = -1(x - 1)

Multiplying both sides by 1, we get:

y - 2 = -x + 1

Rearranging the terms, we get:

x + y = 3

Applications of the Point-Slope Form

The point-slope form of the equation of a straight line has many applications in mathematics and science. Some of the applications include:

Graphing: The point-slope form can be used to graph a straight line when we know a point on the line and the gradient.
Equations of Circles: The point-slope form can be used to derive the equation of a circle when we know the center and the radius.
Equations of Ellipses: The point-slope form can be used to derive the equation of an ellipse when we know the center and the semi-major and semi-minor axes.

Conclusion

In this article, we have shown how to find the equation of a straight line through a given point with a specified gradient using the point-slope form of the equation of a straight line. We have used the point (2, 3) and the gradient -2/3 as an example, and have derived the equation of the line in the standard form. The point-slope form has many applications in mathematics and science, and is a useful tool for graphing and deriving equations of curves.

Introduction

In the previous article, we discussed how to find the equation of a straight line through a given point with a specified gradient using the point-slope form of the equation of a straight line. In this article, we will answer some frequently asked questions about finding the equation of a straight line.

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

A: The point-slope form of the equation of a straight line is given by:

y - y1 = m(x - x1)

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

Q: How do I find the equation of a straight line when I know a point on the line and the gradient?

A: To find the equation of a straight line when you know a point on the line and the gradient, you can use the point-slope form of the equation of a straight line. Simply substitute the values of the point and the gradient into the point-slope form, and simplify the equation.

Q: What if I don't know the gradient of the line?

A: If you don't know the gradient of the line, you can use the slope-intercept form of the equation of a straight line, which is given by:

y = mx + b

where m is the gradient of the line, and b is the y-intercept.

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

A: To find the y-intercept of a straight line, you can use the point-slope form of the equation of a straight line. Simply substitute the values of the point and the gradient into the point-slope form, and simplify the equation. The y-intercept is the value of b in the equation.

Q: Can I use the point-slope form to find the equation of a line when I know two points on the line?

A: Yes, you can use the point-slope form to find the equation of a line when you know two points on the line. Simply substitute the values of the two points into the point-slope form, and simplify the equation.

Q: How do I determine the gradient of a line when I know two points on the line?

A: To determine the gradient of a line when you know two points on the line, you can use the formula:

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

where (x1, y1) and (x2, y2) are the two points on the line.

Q: Can I use the point-slope form to find the equation of a line when I know a point on the line and the slope of the line?

A: Yes, you can use the point-slope form to find the equation of a line when you know a point on the line and the slope of the line. Simply substitute the values of the point and the slope into the point-slope form, and simplify the equation.

Q: How do I find the equation of a line when I know the slope of the line and the y-intercept?

A: To find the equation of a line when you know the slope of the line and the y-intercept, you can use the slope-intercept form of the equation of a straight line, which is given by:

y = mx + b

where m is the slope of the line, and b is the y-intercept.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of a straight line. We have discussed how to use the point-slope form and the slope-intercept form to find the equation of a line when you know a point on the line and the gradient, or when you know two points on the line. We have also discussed how to determine the gradient of a line when you know two points on the line, and how to find the equation of a line when you know the slope of the line and the y-intercept.