Using the Given Point to Find the Equation

Oct 13, 2025 by windailys.com 105 views

Introduction

In mathematics, a straight line is a fundamental concept that can be described using various methods, including the slope-intercept form. The slope-intercept form of a line is given by the equation y = mx + c, where m is the gradient (or slope) of the line, and c is the y-intercept. In this article, we will explore how to find the equation of a straight line when the gradient and a point on the line are given.

Understanding the Problem

We are given a straight line with a gradient of 5 and passing through the point (6, 37). Our goal is to find the equation of this line in the form y = mx + c.

Recalling the Slope-Intercept Form

The slope-intercept form of a line is given by the equation y = mx + c, where m is the gradient (or slope) of the line, and c is the y-intercept. The gradient (m) represents the rate of change of the line, while the y-intercept (c) represents the point at which the line intersects the y-axis.

Using the Given Point to Find the Equation

We are given a point (6, 37) that lies on the line. We can use this point to find the equation of the line. To do this, we can substitute the x and y values of the point into the slope-intercept form of the equation.

Let's substitute x = 6 and y = 37 into the equation y = mx + c:

37 = 5(6) + c

Solving for c

Now, we need to solve for c. To do this, we can simplify the equation:

37 = 30 + c

Subtracting 30 from both sides gives us:

7 = c

Finding the Equation of the Line

Now that we have found the value of c, we can substitute it back into the slope-intercept form of the equation:

y = 5x + 7

This is the equation of the line in the form y = mx + c.

Conclusion

In this article, we have explored how to find the equation of a straight line when the gradient and a point on the line are given. We used the slope-intercept form of the equation and substituted the x and y values of the given point to find the equation of the line. The equation of the line is y = 5x + 7.

Example Use Cases

Finding the Equation of a Line: Given a line with a gradient of 3 and passing through the point (2, 11), find the equation of the line.
Graphing a Line: Given the equation of a line y = 2x + 5, graph the line on a coordinate plane.
Finding the Gradient of a Line: Given the equation of a line y = 4x - 2, find the gradient of the line.

Tips and Tricks

Using the Slope-Intercept Form: The slope-intercept form of a line is a useful tool for finding the equation of a line when the gradient and a point on the line are given.
Substituting Values: When substituting values into the slope-intercept form of the equation, make sure to use the correct values for x and y.
Simplifying Equations: When simplifying equations, make sure to perform the correct operations to isolate the variable.

Common Mistakes

Incorrect Substitution: Failing to substitute the correct values for x and y into the slope-intercept form of the equation.
Incorrect Simplification: Failing to simplify the equation correctly, leading to an incorrect solution.
Incorrect Equation: Writing an incorrect equation, such as y = mx + c, when the correct equation is y = mx + b.
A Straight Line with a Given Gradient and Point: Q&A =====================================================

Introduction

In our previous article, we explored how to find the equation of a straight line when the gradient and a point on the line are given. In this article, we will answer some frequently asked questions related to this topic.

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

A: The slope-intercept form of a line is given by the equation y = mx + c, where m is the gradient (or slope) of the line, and c is the y-intercept.

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

A: To find the equation of a line when the gradient and a point on the line are given, you can use the slope-intercept form of the equation. Substitute the x and y values of the point into the equation and solve for c.

Q: What is the gradient of a line?

A: The gradient of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

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

A: To calculate the gradient 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 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 0.

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

A: To find the y-intercept of a line, you can substitute x = 0 into the equation of the line and solve for y.

Q: Can I use the slope-intercept form of the equation to find the equation of a line when the gradient and two points on the line are given?

A: Yes, you can use the slope-intercept form of the equation to find the equation of a line when the gradient and two points on the line are given. However, you will need to solve a system of equations to find the value of c.

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

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

Failing to substitute the correct values for x and y into the slope-intercept form of the equation
Failing to simplify the equation correctly
Writing an incorrect equation

Q: How do I graph a line on a coordinate plane?

A: To graph a line on a coordinate plane, you can use the equation of the line and plot points on the plane. You can also use the slope-intercept form of the equation to find the y-intercept and plot the line.

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

A: Yes, you can use the slope-intercept form of the equation to find the equation of a line when the y-intercept and a point on the line are given. Simply substitute the x and y values of the point into the equation and solve for m.

Conclusion

In this article, we have answered some frequently asked questions related to finding the equation of a straight line when the gradient and a point on the line are given. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.