Write An Equation Of The Line With The Given Slope, { M$}$, And { Y$}$-intercept { (0, B)$} . . . { M=-\frac{1}{4}, \, B=\frac{1}{2}\$} The Equation Is { \square$}$.(Simplify Your Answer. Type Your Answer In
Introduction
In mathematics, the equation of a line can be represented in various forms, including the slope-intercept form, which is one of the most commonly used forms. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will discuss how to write an equation of a line with a given slope and y-intercept.
Understanding the Slope and Y-Intercept
The slope 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. The y-intercept of a line is the point where the line intersects the y-axis. In other words, it is the value of y when x is equal to 0.
Given Slope and Y-Intercept
In this problem, we are given the slope (m) and y-intercept (b) of a line. The slope is given as m = -1/4, and the y-intercept is given as b = 1/2. We need to write an equation of the line using the slope-intercept form.
Writing the Equation of the Line
To write the equation of the line, we can use the slope-intercept form of the equation, which is y = mx + b. We are given the values of m and b, so we can substitute these values into the equation.
y = (-1/4)x + (1/2)
Simplifying the Equation
The equation we obtained is already in its simplest form. However, we can simplify it further by multiplying both sides of the equation by 4 to eliminate the fractions.
4y = -x + 2
Final Answer
The final answer is 4y = -x + 2.
Conclusion
In this article, we discussed how to write an equation of a line with a given slope and y-intercept. We used the slope-intercept form of the equation, which is y = mx + b, and substituted the given values of m and b into the equation. We also simplified the equation to its final form. This article provides a step-by-step guide on how to write an equation of a line with a given slope and y-intercept.
Example Problems
- Write an equation of a line with a slope of 2/3 and a y-intercept of 1.
- Write an equation of a line with a slope of -3/4 and a y-intercept of 2.
- Write an equation of a line with a slope of 1/2 and a y-intercept of 3.
Solution to Example Problems
- y = (2/3)x + 1
- y = (-3/4)x + 2
- y = (1/2)x + 3
Tips and Tricks
- When writing an equation of a line, make sure to use the correct form of the equation, which is y = mx + b.
- When substituting values into the equation, make sure to use the correct values for m and b.
- When simplifying the equation, make sure to eliminate any fractions by multiplying both sides of the equation by the least common multiple of the denominators.
Common Mistakes
- Using the wrong form of the equation, such as y = mx + c instead of y = mx + b.
- Substituting the wrong values for m and b.
- Failing to simplify the equation to its final form.
Real-World Applications
The equation of a line has many real-world applications, including:
- Calculating the cost of goods sold
- Determining the amount of interest paid on a loan
- Calculating the area of a rectangle
- Determining the height of a building
Conclusion
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is a mathematical equation that represents a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: What is the slope of a line?
A: The slope 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: 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. In other words, it is the value of y when x is equal to 0.
Q: How do I write an equation of a line with a given slope and y-intercept?
A: To write an equation of a line with a given slope and y-intercept, you can use the slope-intercept form of the equation, which is y = mx + b. Simply substitute the given values of m and b into the equation.
Q: What if I have a negative slope? How do I write the equation?
A: If you have a negative slope, you can write the equation by using a negative value for m. For example, if the slope is -1/4, you can write the equation as y = (-1/4)x + b.
Q: Can I use a decimal value for the slope?
A: Yes, you can use a decimal value for the slope. For example, if the slope is 0.5, you can write the equation as y = 0.5x + b.
Q: How do I simplify the equation?
A: To simplify the equation, you can multiply both sides of the equation by the least common multiple of the denominators to eliminate any fractions.
Q: What if I have a fraction for the y-intercept? How do I write the equation?
A: If you have a fraction for the y-intercept, you can write the equation by using the fraction as the value of b. For example, if the y-intercept is 1/2, you can write the equation as y = mx + (1/2).
Q: Can I use a negative value for the y-intercept?
A: Yes, you can use a negative value for the y-intercept. For example, if the y-intercept is -1/2, you can write the equation as y = mx - (1/2).
Q: How do I determine the equation of a line with a given slope and y-intercept?
A: To determine the equation of a line with a given slope and y-intercept, you can use the slope-intercept form of the equation, which is y = mx + b. Simply substitute the given values of m and b into the equation and simplify the result.
Q: What are some real-world applications of the equation of a line?
A: The equation of a line has many real-world applications, including calculating the cost of goods sold, determining the amount of interest paid on a loan, calculating the area of a rectangle, and determining the height of a building.
Q: Can I use the equation of a line to solve problems in other areas of mathematics?
A: Yes, you can use the equation of a line to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.
Q: How do I use the equation of a line to solve problems in real-world applications?
A: To use the equation of a line to solve problems in real-world applications, you can substitute the given values into the equation and solve for the unknown variable. For example, if you are calculating the cost of goods sold, you can substitute the given values into the equation and solve for the total cost.
Conclusion
In conclusion, writing an equation of a line with a given slope and y-intercept is a simple process that involves using the slope-intercept form of the equation and substituting the given values of m and b into the equation. This article provides a step-by-step guide on how to write an equation of a line with a given slope and y-intercept, as well as answers to frequently asked questions about the topic.