I am trying to work through the following equations and can't remember how to do them. I would appreciate if someone could show me how to do both questions so I can see the difference.
Find the equation of the line that passes through
A(4,-2) and is parallel to 2x-3y=7
A(-2,5) and is perpendicular to 2x-y=6
You should try these your self. There are three steps. Do them both the same way.
Put the equation of the line that you want to be parallel to in the form
y = mx + b
That tells you the slope, m.
For the first equation y = (2/3)x -7/3. Therefore the slope is m = 2/3 for that line.
For the second step, write this equation for a line going through the point (x*, y*):
y - y* = m(x - x*)
I find this form very easy to remember. It is easy to see why y must equal y* when x = x*, forcing the line to go through the point.
Then, if you want, rearrange the equation into standard y = mx + b form.
y = mx + (y*-mx*)
Thanks for the help. I tried to work is out and came up with this working out:
A(4,-2) is parallel to 2x-3y=7
y = mx + c
-3y = -2x + 7
y = 2/3x - 2 1/3
(y-y*) / (x-x*) = m
(y+2) / (x-4) = 2/3
y+2 = 2/3(x-4)
y+2 = 2/3x - 2 2/3
y = 2/3x-2 2/3 - 2
y = 2/3x-4 2/3
That is the correct answer. Good work! Now do the other one
To find the equation of a line that passes through a given point and is parallel or perpendicular to another line, we need to understand the relationship between the slopes of the lines.
For a line with equation Ax + By = C, the slope is given by -A/B. This means that two lines are parallel if and only if they have the same slope, and they are perpendicular if and only if the product of their slopes is -1.
Let's work through the first question:
1. Find the slope of the given line 2x - 3y = 7:
Rearrange the equation into slope-intercept form (y = mx + b):
-3y = -2x + 7
Divide by -3 to isolate y:
y = (2/3)x - 7/3
The slope of this line is 2/3.
2. Since we are looking for a line parallel to this one, we know they will have the same slope. So the slope of the parallel line is also 2/3.
3. Use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. We'll use the given point A(4, -2):
y - (-2) = (2/3)(x - 4)
Simplify and rearrange the equation:
y + 2 = (2/3)x - 8/3
Subtract 2 from both sides:
y = (2/3)x - 14/3
Thus, the equation of the line parallel to 2x - 3y = 7 and passing through A(4, -2) is y = (2/3)x - 14/3.
Now let's move on to the second question:
1. Find the slope of the given line 2x - y = 6:
Rearrange the equation into slope-intercept form (y = mx + b):
-y = -2x + 6
Multiply by -1 to isolate y:
y = 2x - 6
The slope of this line is 2.
2. Since we are looking for a line perpendicular to this one, we need to find the negative reciprocal of the slope, which is -1/2.
3. Use the point-slope form of a line, which is y - y1 = m(x - x1). We'll use the given point A(-2, 5):
y - 5 = (-1/2)(x - (-2))
Simplify and rearrange the equation:
y - 5 = (-1/2)x - 1
Add 5 to both sides:
y = (-1/2)x + 4
Thus, the equation of the line perpendicular to 2x - y = 6 and passing through A(-2, 5) is y = (-1/2)x + 4.
By understanding the relationships between slopes and using the point-slope form of a line, we can find the equations of lines parallel or perpendicular to a given line.