Identify An Equation In Slope-intercept Form For The Line Parallel To $y = 5x + 2$ That Passes Through \[$(-6, -1)\$\].A. $y = \frac{1}{5}x + \frac{1}{6}$B. $y = 5x - 29$C. $y = -5x - 11$D. $y = 5x + 29$
Introduction
In mathematics, the slope-intercept form of a linear equation is a fundamental concept used to represent lines on a coordinate plane. The slope-intercept form is given by the equation y = mx + b, where m represents the slope of the line and b is the y-intercept. In this article, we will explore how to identify an equation in slope-intercept form for a line parallel to a given line that passes through a specific point.
Understanding Slope and Parallel Lines
The slope 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. Two lines are parallel if they have the same slope but different y-intercepts. In other words, parallel lines never intersect and have the same steepness.
Given Information
We are given a line in slope-intercept form, y = 5x + 2, and a point (-6, -1) through which a parallel line passes. Our goal is to find the equation of the parallel line that passes through the given point.
Step 1: Identify the Slope of the Given Line
The slope of the given line is 5, which can be determined from the coefficient of x in the slope-intercept form of the equation.
Step 2: Use the Point-Slope Form to Find the Equation of the Parallel Line
The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. We can use this form to find the equation of the parallel line that passes through the given point (-6, -1).
y - (-1) = 5(x - (-6)) y + 1 = 5(x + 6) y + 1 = 5x + 30 y = 5x + 29
Conclusion
In this article, we identified an equation in slope-intercept form for a line parallel to y = 5x + 2 that passes through (-6, -1). The equation of the parallel line is y = 5x + 29. This equation represents a line with the same slope as the given line but a different y-intercept.
Comparison of Options
Let's compare our derived equation with the given options:
A. y = \frac{1}{5}x + \frac{1}{6} - This equation has a different slope and y-intercept, so it is not the correct answer.
B. y = 5x - 29 - This equation has the same slope but a different y-intercept, so it is not the correct answer.
C. y = -5x - 11 - This equation has a different slope, so it is not the correct answer.
D. y = 5x + 29 - This equation has the same slope and y-intercept as our derived equation, so it is the correct answer.
Final Answer
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is given by the equation y = mx + b, where m represents the slope of the line and b is the y-intercept.
Q: What is the significance of the slope in a linear equation?
A: The slope 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. Two lines are parallel if they have the same slope but different y-intercepts.
Q: How do I determine the slope of a line from its equation?
A: To determine the slope of a line from its equation, you need to look at the coefficient of x in the slope-intercept form of the equation. The coefficient of x represents the slope of the line.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: How do I use the point-slope form to find the equation of a parallel line?
A: To use the point-slope form to find the equation of a parallel line, you need to substitute the slope of the given line and the coordinates of the point through which the parallel line passes into the point-slope form of the equation.
Q: What is the relationship between the slope and the y-intercept of parallel lines?
A: The slope of parallel lines is the same, but the y-intercept is different.
Q: How do I determine the equation of a parallel line that passes through a given point?
A: To determine the equation of a parallel line that passes through a given point, you need to use the point-slope form of the equation and substitute the slope of the given line and the coordinates of the point into the equation.
Q: What is the final answer to the problem of identifying an equation in slope-intercept form for a line parallel to y = 5x + 2 that passes through (-6, -1)?
A: The final answer is D. y = 5x + 29.
Q: What are some common mistakes to avoid when identifying equations in slope-intercept form for parallel lines?
A: Some common mistakes to avoid when identifying equations in slope-intercept form for parallel lines include:
- Not using the correct slope of the given line
- Not using the correct coordinates of the point through which the parallel line passes
- Not substituting the correct values into the point-slope form of the equation
- Not simplifying the equation correctly
Q: How can I practice identifying equations in slope-intercept form for parallel lines?
A: You can practice identifying equations in slope-intercept form for parallel lines by working through example problems and exercises. You can also use online resources and practice tests to help you prepare for exams and assessments.
Q: What are some real-world applications of identifying equations in slope-intercept form for parallel lines?
A: Some real-world applications of identifying equations in slope-intercept form for parallel lines include:
- Calculating the cost of materials for a construction project
- Determining the height of a building or a structure
- Calculating the distance between two points on a map
- Determining the slope of a road or a hill
Conclusion
In this article, we have answered some frequently asked questions on identifying equations in slope-intercept form for parallel lines. We have covered topics such as the slope-intercept form of a linear equation, the significance of the slope, and how to use the point-slope form to find the equation of a parallel line. We have also provided some common mistakes to avoid and some real-world applications of identifying equations in slope-intercept form for parallel lines.