For the pairs of lines defined by the following equations indicate with an "I" if they are identical, a "P" if they are distinct but parallel, an "N" (for "normal") if they are perpendicular, and a "G" (for "general") if they are neither parallel nor perpendicular.
3x+4y+5=0 and 6x+8y+10=0.
3x+4y+5=0 and 3x+4y+7=0.
3x+4y+5=0 and 3x+5y+7=0.
−4x+3y+5=0 and 3x+4y+7=0.
Sure, let me analyze each pair of lines for you:
1. 3x+4y+5=0 and 6x+8y+10=0.
- These lines are identical since both equations represent the same line. So, I (identical).
2. 3x+4y+5=0 and 3x+4y+7=0.
- These lines are distinct but parallel since they have the same slope but different y-intercepts. So, P (parallel).
3. 3x+4y+5=0 and 3x+5y+7=0.
- These lines are neither parallel nor perpendicular to each other. They don't have the same slope or the negative reciprocal slopes. So, G (general).
4. -4x+3y+5=0 and 3x+4y+7=0.
- These lines are perpendicular to each other since the product of their slopes is -1. So, N (perpendicular).
Hope that helps! Let me know if there's anything else I can assist you with.
To determine the relationship between the pairs of lines defined by the given equations, we need to identify if they are identical, parallel, perpendicular, or general.
1. 3x+4y+5=0 and 6x+8y+10=0:
Both equations can be simplified by dividing by 2 to obtain:
3x+4y+5=0 (1)
3x+4y+5=0 (2)
Since both equations are the same, they are identical. Therefore, the answer is "I".
2. 3x+4y+5=0 and 3x+4y+7=0:
The equations have the same x and y coefficients but different constants. They are distinct, but the coefficients are the same, indicating that the lines are parallel. Therefore, the answer is "P".
3. 3x+4y+5=0 and 3x+5y+7=0:
The coefficients of x and y are not the same for both equations, which means the lines are not parallel. To determine if they are perpendicular, we need to check if the product of their slopes is -1.
The equation of the first line (1) can be rewritten as:
4y = -3x - 5
Divide by 4 to get: y = (-3/4)x - (5/4)
The equation of the second line (2) can be rewritten as:
5y = -3x - 7
Divide by 5 to get: y = (-3/5)x - (7/5)
The slopes of the two lines are (-3/4) and (-3/5), respectively.
(-3/4) * (-3/5) = 9/20
Since the product is not -1, the lines are not perpendicular. Therefore, the answer is "G".
4. -4x+3y+5=0 and 3x+4y+7=0:
The coefficients of x and y in the two equations are different, indicating that the lines are not parallel. To check if they are perpendicular, we need to calculate the product of their slopes.
The equation of the first line (1) can be rewritten as:
3y = 4x - 5
Divide by 3 to get: y = (4/3)x - (5/3)
The equation of the second line (2) can be rewritten as:
4y = -3x - 7
Divide by 4 to get: y = (-3/4)x - (7/4)
The slopes of the two lines are (4/3) and (-3/4), respectively.
(4/3) * (-3/4) = -1
Since the product is -1, the lines are perpendicular. Therefore, the answer is "N".
To determine the relationship between pairs of lines, we can look at their slope-intercept form, y = mx + b, where "m" represents the slope and "b" represents the y-intercept. If two lines have the same slope (m1 = m2) and different y-intercepts (b1 ≠ b2), they are distinct but parallel. If two lines have slopes that are negative reciprocals of each other (m1 = -1/m2), they are perpendicular. If two lines have different slopes and different y-intercepts, they are neither parallel nor perpendicular.
Let's analyze each pair of lines using this information:
1. 3x + 4y + 5 = 0 and 6x + 8y + 10 = 0:
- First, we need to rewrite the equations in slope-intercept form.
- Simplify the equations by isolating y:
For the first equation: 4y = -3x - 5, divide by 4: y = -3/4x - 5/4.
For the second equation: 8y = -6x - 10, divide by 8: y = -3/4x - 5/4.
- Comparing the equations, we can see that the slopes (m) of both lines are the same (-3/4) and the y-intercepts (b) are also the same (-5/4).
- Therefore, the lines are identical. Answer: I (Identical).
2. 3x + 4y + 5 = 0 and 3x + 4y + 7 = 0:
- Rewrite the equations in slope-intercept form:
For the first equation: 4y = -3x - 5, divide by 4: y = -3/4x - 5/4.
For the second equation: 4y = -3x - 7, divide by 4: y = -3/4x - 7/4.
- Comparing the equations, we can see that the slopes (m) of both lines are the same (-3/4), but the y-intercepts (b) are different.
- Therefore, the lines are distinct but parallel. Answer: P (Distinct but parallel).
3. 3x + 4y + 5 = 0 and 3x + 5y + 7 = 0:
- Rewrite the equations in slope-intercept form:
For the first equation: 4y = -3x - 5, divide by 4: y = -3/4x - 5/4.
For the second equation: 5y = -3x - 7, divide by 5: y = -3/5x - 7/5.
- Comparing the equations, we can see that the slopes (m) are different (-3/4 and -3/5).
- Therefore, the lines are neither parallel nor perpendicular. Answer: G (General).
4. -4x + 3y + 5 = 0 and 3x + 4y + 7 = 0:
- Rewrite the equations in slope-intercept form:
For the first equation: 3y = 4x - 5, divide by 3: y = 4/3x - 5/3.
For the second equation: 4y = -3x - 7, divide by 4: y = -3/4x - 7/4.
- Comparing the equations, we can see that the slopes (m) are negative reciprocals of each other (-3/4 and -4/3).
- Therefore, the lines are perpendicular. Answer: N (Perpendicular).
By analyzing the equations and applying the concepts of slope and intercept, we can determine the relationship between pairs of lines.
the slope of the line Ax+By+C=0 is -A/B
so, check the pairs.
#1 -3/4 vs -6/8 same, so parallel or identical
since multiplying the 1st by 2 gives the 2nd exactly, they are in fact the same line.
do the others in like wise.