Parallel Or Perpendicular Lines Y=3x-7 And 3x+9y=9
Are the lines y=3x-7 and 3x+9y=9 parallel or perpendicular? Explain your answer.
In the realm of mathematics, particularly in coordinate geometry, understanding the relationships between lines is a fundamental concept. Determining whether two lines are parallel or perpendicular is a common task, and it requires a solid grasp of their slopes. This article delves into the specific problem of analyzing the relationship between the lines y = 3x - 7 and 3x + 9y = 9. We will explore the underlying principles, step-by-step calculations, and the reasoning behind the conclusion. Whether you're a student grappling with linear equations or simply seeking to refresh your understanding, this guide will provide a clear and detailed explanation.
Understanding Parallel and Perpendicular Lines
Before diving into the specific equations, let's establish a clear understanding of what it means for lines to be parallel or perpendicular. This foundational knowledge is crucial for accurately analyzing the given problem.
Parallel Lines
Parallel lines are lines that lie in the same plane and never intersect. A key characteristic of parallel lines is that they have the same slope. The slope of a line, often denoted by m, represents the steepness and direction of the line. It is defined as the change in y divided by the change in x (rise over run). If two lines have the same slope, they increase or decrease at the same rate, ensuring they never converge or diverge. In addition to having the same slope, parallel lines must also have different y-intercepts. If they had the same y-intercept, they would be the same line, not parallel lines.
- Key Characteristics of Parallel Lines:
- Same slope (m) : Parallel lines have equal slopes, indicating they have the same steepness and direction.
- Different y-intercepts (b) : Parallel lines must have distinct y-intercepts to avoid being the exact same line.
Perpendicular Lines
Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees). The relationship between the slopes of perpendicular lines is quite specific: their slopes are negative reciprocals of each other. If one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This inverse relationship ensures that the lines intersect at a perfect right angle. Visualizing this, one line slopes upwards to the right, the other downwards, creating the 90-degree intersection.
- Key Characteristics of Perpendicular Lines:
- Negative reciprocal slopes : The product of the slopes of two perpendicular lines equals -1. If one line has a slope of m, the slope of the perpendicular line is -1/m.
- Intersection at a right angle : Perpendicular lines intersect precisely at a 90-degree angle, forming a square corner.
Analyzing the Given Equations
Now that we've established the fundamental principles, let's apply them to the specific problem at hand. We are given two linear equations: y = 3x - 7 and 3x + 9y = 9. Our goal is to determine whether these lines are parallel, perpendicular, or neither. To do this, we need to find the slopes of both lines and then compare them. This involves manipulating the equations into a form where the slope is easily identifiable.
Finding the Slope of the First Line
The first equation, y = 3x - 7, is already in slope-intercept form. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. This form is incredibly convenient because the slope can be directly read off as the coefficient of the x term.
In the equation y = 3x - 7, the coefficient of x is 3. Therefore, the slope of the first line is m1 = 3. This means that for every one unit increase in x, the y value increases by three units. The line slopes upwards quite steeply from left to right.
- Slope-Intercept Form: The equation y = 3x - 7 is in the form y = mx + b, where m is the slope and b is the y-intercept.
- Slope Identification: By comparing the equation with the slope-intercept form, we identify the slope of the first line as m1 = 3.
Finding the Slope of the Second Line
The second equation, 3x + 9y = 9, is not in slope-intercept form. It is in standard form, which is Ax + By = C. To find the slope, we need to rearrange the equation into slope-intercept form (y = mx + b). This involves isolating y on one side of the equation.
To isolate y, we first subtract 3x from both sides of the equation:
3x + 9y - 3x = 9 - 3x
This simplifies to:
9y = -3x + 9
Next, we divide both sides of the equation by 9 to solve for y:
(9y) / 9 = (-3x + 9) / 9
This gives us:
y = (-3/9)x + (9/9)
Simplifying the fractions, we get:
y = (-1/3)x + 1
Now the equation is in slope-intercept form. The coefficient of x is -1/3, so the slope of the second line is m2 = -1/3. This line slopes downwards from left to right, but not as steeply as the first line.
- Standard Form: The equation 3x + 9y = 9 is in the standard form Ax + By = C.
- Conversion to Slope-Intercept Form: To find the slope, we rearranged the equation to y = (-1/3)x + 1.
- Slope Identification: From the slope-intercept form, we identify the slope of the second line as m2 = -1/3.
Determining the Relationship Between the Lines
Having found the slopes of both lines, we can now determine their relationship. The slope of the first line is m1 = 3, and the slope of the second line is m2 = -1/3. We need to check if these slopes satisfy the conditions for parallel or perpendicular lines.
Checking for Parallel Lines
For the lines to be parallel, their slopes must be equal. In this case, m1 = 3 and m2 = -1/3. Clearly, these slopes are not equal, so the lines are not parallel. They do not have the same steepness and direction, and they will eventually intersect if extended indefinitely.
- Slope Comparison: The slopes m1 = 3 and m2 = -1/3 are not equal.
- Conclusion: The lines are not parallel because they have different slopes.
Checking for Perpendicular Lines
For the lines to be perpendicular, their slopes must be negative reciprocals of each other. This means that the product of their slopes must be -1. Let's multiply the slopes to see if this condition is met:
m1 * m2 = 3 * (-1/3) = -1
The product of the slopes is indeed -1. This confirms that the slopes are negative reciprocals of each other. Therefore, the lines are perpendicular.
- Negative Reciprocal Check: We multiplied the slopes m1 = 3 and m2 = -1/3 to get -1.
- Conclusion: The lines are perpendicular because their slopes are negative reciprocals of each other.
Final Answer
Based on our analysis, the lines y = 3x - 7 and 3x + 9y = 9 are perpendicular. We determined this by finding the slopes of both lines and verifying that they are negative reciprocals of each other. This detailed explanation clarifies the process of analyzing the relationship between lines and reinforces the key concepts of slope, parallel lines, and perpendicular lines. This problem illustrates how understanding fundamental mathematical principles allows us to solve complex geometric questions effectively. The ability to manipulate equations, identify slopes, and apply the rules governing line relationships is crucial for success in higher-level mathematics and related fields.