Work Out The Equation Of The Line Which Has A Gradient Of 3 And Passes Through The Point (-1, 3).

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Introduction

In mathematics, the equation of a line can be determined using the gradient-intercept form, which is given by the equation y = mx + c, where m is the gradient of the line and c is the y-intercept. In this article, we will work out the equation of a line that has a gradient of 3 and passes through the point (-1, 3).

Understanding the Gradient

The gradient of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this case, the gradient of the line is given as 3, which means that for every unit of horizontal change, the line will rise by 3 units.

Using the Point-Gradient Form

To find the equation of the line, we can use the point-gradient form, which is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the gradient. In this case, we are given the point (-1, 3) and the gradient (m) is 3.

Substituting the Values

We can now substitute the values into the point-gradient form equation:

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

Simplifying the Equation

To simplify the equation, we can expand the brackets and combine like terms:

y - 3 = 3(x + 1)

y - 3 = 3x + 3

Rearranging the Equation

To put the equation into the standard form (y = mx + c), we can add 3 to both sides of the equation:

y = 3x + 3 + 3

y = 3x + 6

Conclusion

In this article, we have worked out the equation of a line that has a gradient of 3 and passes through the point (-1, 3). We used the point-gradient form to find the equation, and then simplified and rearranged it to put it into the standard form. The final equation of the line is y = 3x + 6.

Examples and Applications

The equation of a line can be used in a variety of real-world applications, such as:

  • Linear Regression: The equation of a line can be used to model the relationship between two variables, such as the price of a product and its demand.
  • Physics: The equation of a line can be used to describe the motion of an object, such as the trajectory of a projectile.
  • Computer Graphics: The equation of a line can be used to create 2D and 3D graphics, such as the lines that make up a 3D model.

Further Reading

For further reading on the equation of a line, we recommend the following resources:

  • Math Open Reference: A comprehensive online reference for mathematics, including the equation of a line.
  • Khan Academy: A free online learning platform that includes video lectures and practice exercises on the equation of a line.
  • Wolfram MathWorld: A comprehensive online encyclopedia of mathematics, including the equation of a line.

Common Mistakes

When working out the equation of a line, it is common to make mistakes such as:

  • Forgetting to add or subtract the gradient: Make sure to add or subtract the gradient correctly when using the point-gradient form.
  • Not simplifying the equation: Make sure to simplify the equation by combining like terms and rearranging it into the standard form.
  • Not checking the solution: Make sure to check the solution by plugging it back into the original equation.

Conclusion

In conclusion, the equation of a line can be determined using the gradient-intercept form, which is given by the equation y = mx + c. In this article, we have worked out the equation of a line that has a gradient of 3 and passes through the point (-1, 3) using the point-gradient form. We have also discussed some common mistakes to avoid when working out the equation of a line.

Introduction

In the previous article, we worked out the equation of a line that has a gradient of 3 and passes through the point (-1, 3). In this article, we will answer some frequently asked questions (FAQs) about the equation of a line.

Q: What is the equation of a line?

A: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of a point on the line. It is typically written in the form y = mx + c, where m is the gradient of the line and c is the y-intercept.

Q: What is the gradient of a line?

A: The gradient of a line is a measure of how steep it 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 find the equation of a line?

A: To find the equation of a line, you can use the point-gradient form, which is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the gradient.

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

A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.

Q: How do I determine the equation of a line if I know two points on the line?

A: To determine the equation of a line if you know two points on the line, you can use the two-point form, which is given by the equation y - y1 = m(x - x1), where (x1, y1) and (x2, y2) are the two points on the line.

Q: What is the difference between the gradient-intercept form and the point-gradient form?

A: The gradient-intercept form is given by the equation y = mx + c, where m is the gradient and c is the y-intercept. The point-gradient form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the gradient.

Q: Can I use the equation of a line to model real-world situations?

A: Yes, the equation of a line can be used to model real-world situations, such as the relationship between two variables, the motion of an object, or the trajectory of a projectile.

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

A: Some common mistakes to avoid when working out the equation of a line include forgetting to add or subtract the gradient, not simplifying the equation, and not checking the solution.

Q: How do I check if my solution is correct?

A: To check if your solution is correct, you can plug it back into the original equation and see if it satisfies the equation.

Q: What are some real-world applications of the equation of a line?

A: Some real-world applications of the equation of a line include linear regression, physics, and computer graphics.

Q: Can I use the equation of a line to solve problems in other areas of mathematics?

A: Yes, the equation of a line can be used to solve problems in other areas of mathematics, such as algebra, geometry, and calculus.

Conclusion

In conclusion, the equation of a line is a fundamental concept in mathematics that can be used to model real-world situations and solve problems in other areas of mathematics. By understanding the equation of a line, you can develop problem-solving skills and apply mathematical concepts to real-world situations.

Further Reading

For further reading on the equation of a line, we recommend the following resources:

  • Math Open Reference: A comprehensive online reference for mathematics, including the equation of a line.
  • Khan Academy: A free online learning platform that includes video lectures and practice exercises on the equation of a line.
  • Wolfram MathWorld: A comprehensive online encyclopedia of mathematics, including the equation of a line.

Common Mistakes

When working out the equation of a line, it is common to make mistakes such as:

  • Forgetting to add or subtract the gradient: Make sure to add or subtract the gradient correctly when using the point-gradient form.
  • Not simplifying the equation: Make sure to simplify the equation by combining like terms and rearranging it into the standard form.
  • Not checking the solution: Make sure to check the solution by plugging it back into the original equation.

Conclusion

In conclusion, the equation of a line is a fundamental concept in mathematics that can be used to model real-world situations and solve problems in other areas of mathematics. By understanding the equation of a line, you can develop problem-solving skills and apply mathematical concepts to real-world situations.