QUESTION 77.1. The Lines Y = 3 4 X − 1 Y = \frac{3}{4}x - 1 Y = 4 3 X − 1 And 4 X + 3 Y = 6 4x + 3y = 6 4 X + 3 Y = 6 Are (parallel Or Perpendicular) To One Another. (Choose The Correct Word.) (1)
Introduction
In mathematics, the relationship between two lines can be either parallel or perpendicular. To determine the relationship between two lines, we need to examine their slopes. In this article, we will explore how to determine if two lines are parallel or perpendicular by analyzing their slopes.
Understanding Slope
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. The slope of a line can be represented by the formula:
m = (y2 - y1) / (x2 - x1)
where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.
Calculating the Slope of the First Line
The first line is represented by the equation y = (3/4)x - 1. To calculate the slope of this line, we can rewrite the equation in the form y = mx + b, where m is the slope and b is the y-intercept.
y = (3/4)x - 1 y = (3/4)x - (1/4)
Comparing this equation to the form y = mx + b, we can see that the slope of the first line is 3/4.
Calculating the Slope of the Second Line
The second line is represented by the equation 4x + 3y = 6. To calculate the slope of this line, we need to rewrite the equation in the form y = mx + b.
3y = -4x + 6 y = (-4/3)x + 2
Comparing this equation to the form y = mx + b, we can see that the slope of the second line is -4/3.
Determining the Relationship Between the Two Lines
Now that we have calculated the slopes of both lines, we can determine their relationship. If the slopes of two lines are equal, then the lines are parallel. If the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.
Checking for Parallel Lines
To check if the two lines are parallel, we need to compare their slopes. If the slopes are equal, then the lines are parallel.
m1 = 3/4 m2 = -4/3
Since the slopes are not equal, the lines are not parallel.
Checking for Perpendicular Lines
To check if the two lines are perpendicular, we need to compare their slopes. If the slopes are negative reciprocals of each other, then the lines are perpendicular.
m1 = 3/4 m2 = -4/3
Since the slopes are not negative reciprocals of each other, the lines are not perpendicular.
Conclusion
In conclusion, the two lines y = (3/4)x - 1 and 4x + 3y = 6 are neither parallel nor perpendicular. The slope of the first line is 3/4, and the slope of the second line is -4/3. Since the slopes are not equal and not negative reciprocals of each other, the lines are neither parallel nor perpendicular.
Final Answer
Q: What is the difference between parallel and perpendicular lines?
A: Parallel lines are lines that never intersect, no matter how far they are extended. Perpendicular lines, on the other hand, are lines that intersect at a 90-degree angle.
Q: How do I determine if two lines are parallel or perpendicular?
A: To determine if two lines are parallel or perpendicular, you need to examine their slopes. If the slopes are equal, then the lines are parallel. If the slopes are negative reciprocals of each other, then the lines are perpendicular.
Q: What is the slope of a line?
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.
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 m is the slope, and (x1, y1) and (x2, y2) are two points on the line.
Q: What is the difference between a slope of 1 and a slope of -1?
A: A slope of 1 represents a line that rises 1 unit for every 1 unit it runs. A slope of -1 represents a line that falls 1 unit for every 1 unit it runs.
Q: Can two lines have the same slope but not be parallel?
A: No, two lines cannot have the same slope but not be parallel. If two lines have the same slope, then they are parallel.
Q: Can two lines have negative reciprocal slopes but not be perpendicular?
A: No, two lines cannot have negative reciprocal slopes but not be perpendicular. If two lines have negative reciprocal slopes, then they are perpendicular.
Q: How do I determine if a line is horizontal or vertical?
A: A horizontal line has a slope of 0, while a vertical line has an undefined slope.
Q: Can a line be both horizontal and vertical at the same time?
A: No, a line cannot be both horizontal and vertical at the same time. A line can be either horizontal or vertical, but not both.
Q: What is the relationship between the slopes of parallel lines?
A: The slopes of parallel lines are equal.
Q: What is the relationship between the slopes of perpendicular lines?
A: The slopes of perpendicular lines are negative reciprocals of each other.
Q: Can two lines have the same y-intercept but not be parallel?
A: No, two lines cannot have the same y-intercept but not be parallel. If two lines have the same y-intercept, then they are parallel.
Q: Can two lines have the same x-intercept but not be perpendicular?
A: No, two lines cannot have the same x-intercept but not be perpendicular. If two lines have the same x-intercept, then they are perpendicular.
Conclusion
In conclusion, understanding the relationship between parallel and perpendicular lines is crucial in mathematics. By examining the slopes of two lines, you can determine if they are parallel, perpendicular, or neither. We hope this FAQ article has provided you with a better understanding of parallel and perpendicular lines.