13. Determine if the lines [tex]y=2 x+5[/tex] and [tex]4 x-2 y=10[/tex] are parallel, perpendicular, or neither.
14. Are the lines [tex]y=-\frac{1}{3} x+4[/tex] and [tex]x-3 y=9[/tex] parallel, perpendicular, or neither?
Answer (1)
For the first pair of lines, rewrite the second equation in slope-intercept form and compare the slopes. Since the slopes are equal, the lines are parallel.
For the second pair of lines, rewrite the second equation in slope-intercept form and compare the slopes. Since the product of the slopes is not -1 and the slopes are not equal, the lines are neither parallel nor perpendicular.
The first pair of lines are parallel.
The second pair of lines are neither parallel nor perpendicular. Parallel, Neither
Explanation
Analyze the first pair of lines. Let's analyze the first pair of lines:
Line 1: y = 2 x + 5 . This is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. So, the slope of Line 1 is m 1 = 2 .
Line 2: 4 x − 2 y = 10 . We need to rewrite this in slope-intercept form to find its slope.
Rewrite Line 2 in slope-intercept form. To rewrite Line 2, we isolate y :
4 x − 2 y = 10
Subtract 4 x from both sides:
− 2 y = − 4 x + 10
Divide both sides by − 2 :
y = 2 x − 5
So, the slope of Line 2 is m 2 = 2 .
Compare the slopes of Line 1 and Line 2. Now, let's compare the slopes of Line 1 and Line 2. We have m 1 = 2 and m 2 = 2 . Since m 1 = m 2 , the lines are parallel.
Analyze the second pair of lines. Now let's analyze the second pair of lines:
Line 3: y = − 3 1 x + 4 . This is in slope-intercept form, so the slope of Line 3 is m 3 = − 3 1 .
Line 4: x − 3 y = 9 . We need to rewrite this in slope-intercept form to find its slope.
Rewrite Line 4 in slope-intercept form. To rewrite Line 4, we isolate y :
x − 3 y = 9
Subtract x from both sides:
− 3 y = − x + 9
Divide both sides by − 3 :
y = 3 1 x − 3
So, the slope of Line 4 is m 4 = 3 1 .
Compare the slopes of Line 3 and Line 4. Now, let's compare the slopes of Line 3 and Line 4. We have m 3 = − 3 1 and m 4 = 3 1 .
To check if the lines are perpendicular, we multiply their slopes:
m 3 × m 4 = − 3 1 × 3 1 = − 9 1
Since the product of the slopes is not − 1 , the lines are not perpendicular. Since the slopes are not equal, the lines are not parallel. Therefore, the lines are neither parallel nor perpendicular.
Examples
Understanding the relationships between lines is crucial in architecture and design. For instance, when designing a building, architects need to ensure that certain walls are parallel for structural integrity or perpendicular for aesthetic purposes. By calculating the slopes of lines representing these walls, they can verify these relationships and ensure the building is designed correctly.
