What Is The Equation Of The Line That Goes Through The Points { (-5,-1)$}$ And { (5,5)$}$?A. ${y=\frac{3}{5} X-8}$B. ${y=\frac{2}{5} X+3}$C. ${y=\frac{3}{5} X+2}$D. ${y=\frac{2}{5} X-7}$
What is the Equation of the Line that Goes Through the Points (-5,-1) and (5,5)?
In mathematics, finding the equation of a line that passes through two given points is a fundamental concept in algebra and geometry. The equation of a line can be expressed in various forms, including the slope-intercept form, point-slope form, and standard form. In this article, we will explore how to find the equation of a line that passes through the points (-5,-1) and (5,5).
To find the equation of the line that passes through the points (-5,-1) and (5,5), we need to use the concept of slope and the point-slope form of a line. The slope of a line is a measure of how steep it is, and it can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where m is the slope, and (x1, y1) and (x2, y2) are the coordinates of the two points.
Using the given points (-5,-1) and (5,5), we can calculate the slope of the line as follows:
m = (5 - (-1)) / (5 - (-5)) m = (5 + 1) / (5 + 5) m = 6 / 10 m = 3/5
Now that we have the slope, we can use the point-slope form of a line to find the equation of the line that passes through the points (-5,-1) and (5,5). The point-slope form of a line is given by:
y - y1 = m(x - x1)
where m is the slope, and (x1, y1) is one of the given points.
Using the point (-5,-1), we can substitute the values into the equation as follows:
y - (-1) = (3/5)(x - (-5)) y + 1 = (3/5)(x + 5)
To simplify the equation, we can multiply both sides by 5 to eliminate the fraction:
5(y + 1) = 3(x + 5) 5y + 5 = 3x + 15
Now, we can subtract 5 from both sides to isolate the term with y:
5y = 3x + 10
Finally, we can divide both sides by 5 to solve for y:
y = (3/5)x + 2
In conclusion, the equation of the line that passes through the points (-5,-1) and (5,5) is y = (3/5)x + 2. This equation represents a line with a slope of 3/5 and a y-intercept of 2.
Now, let's compare our answer with the answer choices provided:
A. y = (3/5)x - 8 B. y = (2/5)x + 3 C. y = (3/5)x + 2 D. y = (2/5)x - 7
Our answer, y = (3/5)x + 2, matches answer choice C.
The final answer is C.
Frequently Asked Questions (FAQs) About Finding the Equation of a Line
In our previous article, we explored how to find the equation of a line that passes through two given points. In this article, we will answer some frequently asked questions (FAQs) about finding the equation of a line.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is a way to express the equation of a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: How do I find the slope of a line?
A: To find 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 point-slope form of a line?
A: The point-slope form of a line is a way to express the equation of a line in the form y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.
Q: How do I find the equation of a line that passes through two points?
A: To find the equation of a line that passes through two points, you can use the point-slope form of a line. First, find the slope of the line using the formula m = (y2 - y1) / (x2 - x1). Then, substitute the values into the point-slope form of a line and simplify the equation.
Q: What is the standard form of a line?
A: The standard form of a line is a way to express the equation of a line in the form Ax + By = C, where A, B, and C are constants.
Q: How do I convert the equation of a line from slope-intercept form to standard form?
A: To convert the equation of a line from slope-intercept form to standard form, you can multiply both sides of the equation by the denominator of the slope and then rearrange the terms.
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 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: What is the x-intercept of a line?
A: The x-intercept of a line is the point where the line intersects the x-axis. It is the value of x when y is equal to 0.
Q: How do I find the x-intercept of a line?
A: To find the x-intercept of a line, you can substitute y = 0 into the equation of the line and solve for x.
In conclusion, finding the equation of a line that passes through two points is a fundamental concept in algebra and geometry. By understanding the slope-intercept form, point-slope form, and standard form of a line, you can find the equation of a line that passes through two points. We hope this article has helped you to understand the concepts and formulas involved in finding the equation of a line.
If you want to learn more about finding the equation of a line, we recommend checking out the following resources:
- Khan Academy: Finding the Equation of a Line
- Mathway: Finding the Equation of a Line
- Wolfram Alpha: Finding the Equation of a Line
We hope this article has been helpful in answering your questions about finding the equation of a line. If you have any further questions, please don't hesitate to ask.