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Tutorial 1.17 Determine whether the lines for a pair of equations are parallel, perpendicular or coincide with each other
When two linear equations are graphed on the same coordinate grid, these lines can interact with each other in one of three ways:
• The two graphed lines can intersect each other…meeting at exactly one point. If the two lines form right angles, they are considered perpendicular with each other…but, the lines will still only have one point of intersection. (See diagram #1 below for an example.)
• The two graphed lines can be parallel with each other…meaning that they do not touch each (intersect with) other at all. (See diagram #2 below for an example.)
• The two graphed lines can coincide…meaning that the lines will lie on top of each other.
Although there will be two equations, the finalized graph will look as if there was only one line drawn. (See diagram #3 below for an example.)
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Here are the equations that produced the graphs above:
5x + 4y = 20 4x – 5y = 20 6x – 4y = 12 3x – 2y = 18 5x – 2y = 10 10x – 4y = 20 slope = –5/4
(0, y) = (0, 5) (x, 0) = (4, 0)
slope = 4/5 (0, y) = (0, –4)
(x, 0) = (5, 0)
slope = 6/4 (0, y) = (0, 3) (x, 0) = (2, 0)
slope = 3/2 (0, y) = (0, –9)
(x, 0) = (6, 0)
slope = 5/2 (0, y) = (0, –5)
(x, 0) = (2, 0)
slope = 10/4 (0, y) = (0, –5)
(x, 0) = (2, 0) We can analyze what elements of the linear equations will produce each of the possible interactions.
Below each of the graphed equations, we have listed the slope, y-intercept and x-intercept for the graphed lines. Notice that:
• The two graphed lines will intersect when the two slopes are not equal (diagram #1).
• The two graphed lines will be perpendicular to each other when the two slopes are negative reciprocals of each other…meaning that the product of the two slopes will be –1. (diagram #1)
51
• The two graphed lines will be parallel to each other when the two slopes are equal to each other but the y-intercepts are not. (diagram #2)
• The two graphed lines will coincide when both the two slopes and the y-intercepts are equal to each other. (diagram #3)
We can use these observations to determine whether two lines will be parallel, perpendicular, coincide or just intersect directly from their equations. Just follow these steps:
1. Find the slope for each given equation.
2. Find the y-intercept for each given equation.
3. Use the information obtained to determine how the lines will interact with each other.
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example 1.17a Determine whether these lines will be parallel, perpendicular, coincide, or just intersect: 5x – 2y = 20
10x – 4y = 20 Follow these steps:
1. Find the slope for each given equation:
5x – 2y = 20 → m = –A/B = –5/–2 = 5/2 = 2.5 10x – 4y = 20 → m = –A/B = –10/–4 = 5/2 = 2.5 2. Find the y-intercept for each given equation:
5x – 2y = 20 → b = C/B = 20/–2 = 10 → (0, –10) 10x – 4y = 20 → b = C/B = 20/–4 = –5 → (0, –5)
3. Use the information obtained to determine how the lines will interact with each other: The slopes are equal but the y-intercepts are not. Thus, the lines will be parallel to each other.
example 1.17b Determine whether these lines will be parallel, perpendicular, coincide, or just intersect: 5x + 4y = 9
y = 4 5x + 9 Follow these steps:
1. Find the slope for each given equation:
5x + 4y = 9 → m = –A/B = –5/4 = –1.25 y = (4/5)x + 9 → m = 4/5 = 0.8
2. Find the y-intercept for each given equation:
5x + 4y = 9 → b = C/B = 9/4 = 2.25 → (0, 2.25) y = (4/5)x + 9 → b = 9 → (0, 9)
3. Use the information obtained to determine how the lines will interact with each other: The slopes are not equal but (–5/4)(4/5) = –1. Thus, the lines will be perpendicular to each other.
example 1.17c Determine whether these lines will be parallel, perpendicular, coincide, or just intersect: 6x + 4y = 24
3x + 2y = 12 Follow these steps:
1. Find the slope for each given equation:
6x + 4y = 24 → m = –A/B = –6/4 = –3/2 = –1.5 3x + 2y = 12 → m = –A/B = –3/2 = –1.5
53 2. Find the y-intercept for each given equation:
6x + 4y = 24 → b = C/B = 24/4 = 6 → (0, 6) 3x + 2y = 12 → b = C/B = 12/2 = 6 → (0, 6)
3. Use the information obtained to determine how the lines will interact with each other: Both the slopes and the y-intercepts are equal. Thus, the lines will coincide.
example 1.17d Determine whether these lines will be parallel, perpendicular, coincide, or just intersect: 3x + 2y = 10
2x – y = 6 Follow these steps:
1. Find the slope for each given equation:
3x + 2y = 10 → m = –A/B = –3/2 = –1.5 2x – y = 6 → m = –A/B = –2/–1 = 2/1 = 2 2. Find the y-intercept for each given equation:
3x + 2y = 10 → b = C/B = 10/2 = 5 → (0, 5) 2x – y = 6 → b = C/B = 6/–1 = –6 → (0, –6)
3. Use the information obtained to determine how the lines will interact with each other: The slopes are not equal and they are not negative reciprocals of each other. Thus, the lines will intersect with each other at one specific point.
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