Work Out The Equation Of The Straight Line That Passes Through The Points { (6,8)$}$ And { (9,23)$}$.Give Your Answer In The Form { Y = Mx + C$}$, Where { M$}$ And { C$}$ Are Integers Or Fractions In
Introduction
In mathematics, the equation of a straight line is a fundamental concept that is used to describe the relationship between two variables, typically x and y. Given two points on a line, we can use the concept of slope and y-intercept to determine the equation of the line. In this article, we will work out the equation of the straight line that passes through the points (6,8) and (9,23).
What is the Equation of a Straight Line?
The equation of a straight line is typically written in the form y = mx + c, where m is the slope of the line and c is the y-intercept. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Calculating the Slope
To calculate the slope of the line that passes through the points (6,8) and (9,23), we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Plugging in the values, we get:
m = (23 - 8) / (9 - 6) m = 15 / 3 m = 5
Calculating the Y-Intercept
Now that we have the slope, we can use one of the points to calculate the y-intercept. We will use the point (6,8).
We know that the equation of the line is y = mx + c, and we have already calculated the slope (m = 5). Plugging in the values, we get:
8 = 5(6) + c 8 = 30 + c c = -22
Writing the Equation of the Line
Now that we have the slope (m = 5) and the y-intercept (c = -22), we can write the equation of the line in the form y = mx + c.
y = 5x - 22
Conclusion
In this article, we worked out the equation of the straight line that passes through the points (6,8) and (9,23). We calculated the slope (m = 5) and the y-intercept (c = -22), and then wrote the equation of the line in the form y = mx + c.
Example Use Cases
The equation of a straight line has many practical applications in real-world problems. Here are a few examples:
- Physics: The equation of a straight line can be used to describe the motion of an object under constant acceleration.
- Engineering: The equation of a straight line can be used to design and optimize systems, such as bridges and buildings.
- Computer Science: The equation of a straight line can be used to implement algorithms for computer graphics and game development.
Tips and Tricks
Here are a few tips and tricks to help you work out the equation of a straight line:
- Use the slope formula: The slope formula (m = (y2 - y1) / (x2 - x1)) is a powerful tool for calculating the slope of a line.
- Choose a point: When calculating the y-intercept, choose a point that is easy to work with.
- Check your work: Always check your work to make sure that the equation of the line is correct.
Common Mistakes
Here are a few common mistakes to avoid when working out the equation of a straight line:
- Incorrect slope: Make sure to calculate the slope correctly using the slope formula.
- Incorrect y-intercept: Make sure to calculate the y-intercept correctly using one of the points.
- Incorrect equation: Make sure to write the equation of the line in the correct form (y = mx + c).
Frequently Asked Questions: Solving the Equation of a Straight Line ====================================================================
Q: What is the equation of a straight line?
A: The equation of a straight line is a mathematical expression that describes the relationship between two variables, typically x and y. It is typically written in the form y = mx + c, where m is the slope of the line and c is the y-intercept.
Q: How do I calculate the slope of a line?
A: To calculate the slope of a line, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of two points on the line.
Q: What is the y-intercept?
A: The y-intercept is the point on the line where the x-coordinate is zero. It is the value of y when x is zero.
Q: How do I calculate the y-intercept?
A: To calculate the y-intercept, you can use one of the points on the line and the slope. For example, if you know the slope (m) and the point (x, y), you can use the equation:
y = mx + c
to solve for c, the y-intercept.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable (x) is 1. For example, y = 2x + 3 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable (x) is 2. For example, y = x^2 + 3x + 2 is a quadratic equation.
Q: Can I use the equation of a straight line to solve problems in physics?
A: Yes, the equation of a straight line can be used to solve problems in physics, such as calculating the motion of an object under constant acceleration.
Q: Can I use the equation of a straight line to solve problems in engineering?
A: Yes, the equation of a straight line can be used to solve problems in engineering, such as designing and optimizing systems, such as bridges and buildings.
Q: What are some common mistakes to avoid when working out the equation of a straight line?
A: Some common mistakes to avoid when working out the equation of a straight line include:
- Incorrect slope
- Incorrect y-intercept
- Incorrect equation
Q: How can I check my work when solving the equation of a straight line?
A: You can check your work by plugging in the values of x and y into the equation and making sure that the equation is true.
Q: What are some real-world applications of the equation of a straight line?
A: Some real-world applications of the equation of a straight line include:
- Physics: calculating the motion of an object under constant acceleration
- Engineering: designing and optimizing systems, such as bridges and buildings
- Computer Science: implementing algorithms for computer graphics and game development
Q: Can I use the equation of a straight line to solve problems in other fields?
A: Yes, the equation of a straight line can be used to solve problems in other fields, such as economics, finance, and social sciences.
Q: What are some tips and tricks for working out the equation of a straight line?
A: Some tips and tricks for working out the equation of a straight line include:
- Using the slope formula
- Choosing a point that is easy to work with
- Checking your work to make sure that the equation is correct