Work Out The Equation Of The Line Which Has A Gradient Of 3 And Passes Through The Point (-1, 3).
Introduction
In mathematics, the equation of a line can be determined using the point-slope form, which is given by the formula: y - y1 = m(x - x1), where m is the gradient of the line and (x1, y1) is a point on the line. In this article, we will work out the equation of the line which has a gradient of 3 and passes through the point (-1, 3).
Understanding the Point-Slope Form
The point-slope form is a fundamental concept in mathematics, and it is used to find the equation of a line when we know its gradient and a point on the line. The formula is given by: y - y1 = m(x - x1), where m is the gradient of the line and (x1, y1) is a point on the line. In this case, we are given that the gradient of the line is 3 and the point on the line is (-1, 3).
Substituting the Values into the Formula
To find the equation of the line, we need to substitute the values of the gradient and the point into the point-slope form. The gradient is given as 3, and the point on the line is (-1, 3). Substituting these values into the formula, we get:
y - 3 = 3(x - (-1))
Simplifying the Equation
To simplify the equation, we need to expand the brackets and combine like terms. Expanding the brackets, we get:
y - 3 = 3(x + 1)
Distributing the Gradient
To distribute the gradient, we need to multiply the gradient by the terms inside the brackets. Multiplying the gradient by the terms inside the brackets, we get:
y - 3 = 3x + 3
Rearranging the Equation
To rearrange the equation, we need to isolate the y-term on one side of the equation. Adding 3 to both sides of the equation, we get:
y = 3x + 6
Conclusion
In this article, we have worked out the equation of the line which has a gradient of 3 and passes through the point (-1, 3). We used the point-slope form to find the equation of the line, and we simplified the equation to get the final result. The equation of the line is y = 3x + 6.
Example
Let's use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is 2. Substituting x = 2 into the equation of the line, we get:
y = 3(2) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 6 + 6
Simplifying the Expression
To simplify the expression, we need to combine like terms. Combining like terms, we get:
y = 12
Conclusion
In this example, we used the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is 2. The y-coordinate of the point is 12.
Graphing the Line
To graph the line, we need to plot the points on a coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value. For example, we can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is 0. Substituting x = 0 into the equation of the line, we get:
y = 3(0) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 6
Plotting the Point
To plot the point, we need to mark the point on the coordinate plane. The point is (0, 6).
Plotting the Line
To plot the line, we need to plot the points on the coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value. For example, we can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is 1. Substituting x = 1 into the equation of the line, we get:
y = 3(1) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 9
Plotting the Point
To plot the point, we need to mark the point on the coordinate plane. The point is (1, 9).
Plotting the Line
To plot the line, we need to plot the points on the coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value. For example, we can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is -1. Substituting x = -1 into the equation of the line, we get:
y = 3(-1) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 3
Plotting the Point
To plot the point, we need to mark the point on the coordinate plane. The point is (-1, 3).
Plotting the Line
To plot the line, we need to plot the points on the coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value. For example, we can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is 2. Substituting x = 2 into the equation of the line, we get:
y = 3(2) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 12
Plotting the Point
To plot the point, we need to mark the point on the coordinate plane. The point is (2, 12).
Plotting the Line
To plot the line, we need to plot the points on the coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value. For example, we can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is 0. Substituting x = 0 into the equation of the line, we get:
y = 3(0) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 6
Plotting the Point
To plot the point, we need to mark the point on the coordinate plane. The point is (0, 6).
Plotting the Line
To plot the line, we need to plot the points on the coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value. For example, we can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is 1. Substituting x = 1 into the equation of the line, we get:
y = 3(1) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 9
Plotting the Point
To plot the point, we need to mark the point on the coordinate plane. The point is (1, 9).
Plotting the Line
To plot the line, we need to plot the points on the coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value. For example, we can use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is -1. Substituting x = -1 into the equation of the line, we get:
y = 3(-1) + 6
Evaluating the Expression
To evaluate the expression, we need to multiply the gradient by the x-coordinate and add the result to 6. Multiplying the gradient by the x-coordinate, we get:
y = 3
Plotting the Point
To plot the point, we need to mark the point on the coordinate plane. The point is (-1, 3).
Plotting the Line
To plot the line, we need to plot the points on the coordinate plane. We can use the equation of the line to find the y-coordinate of a point on the line when the
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 + b, where m is the gradient of the line and b is the y-intercept.
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. In the equation y = mx + b, the gradient is represented by the coefficient m.
Q: How do I find the equation of a line if I know its gradient and a point on the line?
A: To find the equation of a line if you know its gradient and a point on the line, you can use the point-slope form of the equation, which is y - y1 = m(x - x1), where m is the gradient and (x1, y1) is the point on the line.
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. In the equation y = mx + b, the y-intercept is represented by the constant term b.
Q: How do I graph a line if I know its equation?
A: To graph a line if you know its equation, you can use the following steps:
- Identify the y-intercept of the line, which is the point where the line intersects the y-axis.
- Plot the y-intercept on the coordinate plane.
- Use the equation of the line to find the y-coordinate of a point on the line when the x-coordinate is a specific value.
- Plot the point on the coordinate plane.
- Repeat steps 3 and 4 for several points on the line to create a graph of the line.
Q: What is the difference between a linear equation and a non-linear equation?
A: A linear equation is an equation that can be written in the form y = mx + b, where m is the gradient and b is the y-intercept. A non-linear equation is an equation that cannot be written in this form.
Q: Can I use the equation of a line to solve real-world problems?
A: Yes, the equation of a line can be used to solve a wide range of real-world problems, including problems involving finance, science, engineering, and more.
Q: How do I use the equation of a line to solve a problem?
A: To use the equation of a line to solve a problem, you can follow these steps:
- Identify the problem and the variables involved.
- Write an equation that represents the relationship between the variables.
- Use the equation to find the solution to the problem.
Q: What are some common applications of the equation of a line?
A: Some common applications of the equation of a line include:
- Finance: calculating interest rates and investment returns
- Science: modeling the motion of objects and predicting future behavior
- Engineering: designing and building structures and systems
- Economics: analyzing and predicting economic trends and behavior
Q: Can I use the equation of a line to solve problems involving multiple variables?
A: Yes, the equation of a line can be used to solve problems involving multiple variables. However, you will need to use a more complex equation, such as a system of linear equations or a quadratic equation.
Q: How do I use the equation of a line to solve a problem involving multiple variables?
A: To use the equation of a line to solve a problem involving multiple variables, you can follow these steps:
- Identify the problem and the variables involved.
- Write an equation that represents the relationship between the variables.
- Use the equation to find the solution to the problem.
Q: What are some common mistakes to avoid when using the equation of a line?
A: Some common mistakes to avoid when using the equation of a line include:
- Failing to identify the correct equation of the line
- Making errors in the calculation of the gradient or y-intercept
- Failing to consider the domain and range of the line
- Failing to use the correct units and measurements
Q: How do I check my work when using the equation of a line?
A: To check your work when using the equation of a line, you can follow these steps:
- Verify that the equation is in the correct form (y = mx + b)
- Check that the gradient and y-intercept are correct
- Use the equation to find the y-coordinate of a point on the line when the x-coordinate is a specific value
- Plot the point on the coordinate plane to verify that it is on the line
Q: What are some common resources for learning about the equation of a line?
A: Some common resources for learning about the equation of a line include:
- Textbooks and online resources
- Video tutorials and online courses
- Practice problems and worksheets
- Real-world examples and applications
Q: How do I practice using the equation of a line?
A: To practice using the equation of a line, you can follow these steps:
- Practice solving problems involving the equation of a line
- Use online resources and practice problems to reinforce your understanding
- Apply the equation of a line to real-world problems and scenarios
- Review and practice regularly to maintain your skills and knowledge.