please check my homework
given P(7,2) and the line y + 3x=7 find the equation for the line
parallel line: y+3x+7
y=-3(x-7) +7
y-2=-3x+21+7
y=-3x+28+2
y=-3x+30
perpendicular line:
y-2=1/3(x-7) +7
y-2=1/3x-7/3 +21/3
y-2==1/3x + 14
y=1/3x + 14 +2
y=1/3x +16
Here is a neat way to do these question in a quicker way
You should have learned that for a straight line equation in the form
Ax + By = C, the slope of the line is -A/B
so for 3x+y = 7 , a parallel line will differ only in the C value
so new equation is
3x + y = C
plug in the point (7,2)
21 + 2 = C = 23
so new equation: 3x + y = 23 or y = -3x - 23
notice the given point does not even satisfy your final equation.
Your error is in the 2nd line, should be
y = -3(x-7) + 2
For the perpendicular short cut way:
slope of given equation is -3
so slope of perpendicular line is +1/3
new equation:
x - 3y = C, plug in the point (7,2)
7 - 6 = C = 1
x - 3y = 1 or y = (1/3)x - 1/3
again, you made the error in the 2nd line, it should say
y-2 = (1/3)(x-7)
then 3y - 6 = x-7
x - 3y = 1 , the same result as mine
To check your work, let's solve the problem step by step:
Given: P(7,2) and the line y + 3x = 7.
1) For a parallel line, the slope will be the same as the given line. Start by rearranging the given line to the slope-intercept form (y = mx + b):
y + 3x = 7
y = -3x + 7
Since the coefficient of x is -3, the slope (m) of this line is also -3. So the equation for the parallel line has the same slope. Now, substitute the given point P(7,2) into the equation for the parallel line:
y - 2 = -3(x - 7)
Distribute -3:
y - 2 = -3x + 21
Move the constant to the other side:
y = -3x + 21 + 2 = -3x + 23
Therefore, the equation for the parallel line is y = -3x + 23.
2) For a perpendicular line, the slope is the negative reciprocal of the given line's slope. The given line has a slope of -3, so the negative reciprocal of -3 is 1/3. Start by rearranging the given line to the slope-intercept form:
y + 3x = 7
y = -3x + 7
Now, find the negative reciprocal:
-3 x (1/3) = -1
So the slope (m) of the perpendicular line is 1/3. Using the point-slope form, substitute the given point P(7,2) into the equation:
y - 2 = (1/3)(x - 7)
Distribute (1/3):
y - 2 = (1/3)x - (7/3)
Move the constant to the other side:
y = (1/3)x - (7/3) + 2 = (1/3)x - (7/3) + (6/3) = (1/3)x - (1/3)
Therefore, the equation for the perpendicular line is y = (1/3)x - (1/3).
Based on these steps, it appears that your equations for the parallel and perpendicular lines are incorrect.