Write a slope-intercept equation for a line passing through the point (4,−2) that is parallel to the line 4x+7y=8. Then write a second equation for a line passing through the point (4,−2) that is perpendicular to the line 4x+7y=8.
Thank you!
Sure, here you go:
For the line that is parallel to the line 4x + 7y = 8:
To find the slope, we can rearrange the given equation to the slope-intercept form, which is y = mx + b. So let's solve for y:
4x + 7y = 8
7y = -4x + 8
y = (-4/7)x + 8/7
Since the line we're looking for is parallel, it will have the same slope as the given line. Therefore, the slope of the parallel line is -4/7.
Now we can substitute the point (4,-2) and the slope into the slope-intercept form to find the equation:
y = mx + b
-2 = (-4/7) (4) + b
-2 = -16/7 + b
To find the value of b, we can simplify:
-2 = -16/7 + b
-2 + 16/7 = b
-14/7 + 16/7 = b
2/7 = b
Therefore, the equation for the line parallel to 4x + 7y = 8 and passing through (4,-2) is:
y = (-4/7)x + 2/7
Now, let's find the equation for the line that is perpendicular to 4x + 7y = 8:
The slope of the perpendicular line will be the negative reciprocal of the slope of the given line. So the slope of the given line is -4/7, and the slope of the perpendicular line will be 7/4.
Using the point (4,-2) and the slope 7/4, we can find the equation:
y = mx + b
-2 = (7/4)(4) + b
-2 = 7 + b
Solving for b:
-2 - 7 = b
-9 = b
Therefore, the equation for the line perpendicular to 4x + 7y = 8 and passing through (4,-2) is:
y = (7/4)x - 9
I hope that helps! Let me know if you need any more assistance.
To find the equation of a line that is parallel to the given line, we need to find the slope first.
The given equation is 4x+7y=8.
First, rearrange it into slope-intercept form (y = mx + b) by isolating y:
7y = -4x + 8
Divide by 7:
y = -4/7x + 8/7
From this equation, we can see that the slope of the line is -4/7. So, any line parallel to it will have the same slope.
Now, use the point-slope form of a line (y - y1 = m(x - x1)) and substitute the values of the given point (4, -2) and the slope (-4/7) into the formula.
y - (-2) = -4/7(x - 4)
y + 2 = -4/7(x - 4)
y + 2 = -4/7x + 16/7
Rearrange the equation in slope-intercept form:
y = -4/7x + 16/7 - 2
y = -4/7x + 16/7 - 14/7
y = -4/7x + 2/7
So, the equation of a line passing through the point (4, -2) and parallel to 4x+7y=8 is y = -4/7x + 2/7.
To find the equation of a line perpendicular to the given line, we'll need to calculate the negative reciprocal of the slope of 4x+7y=8, which is 4/7.
The negative reciprocal of 4/7 is -7/4.
Using the point-slope form, substitute the values of the given point (4, -2) and the slope (-7/4) into the formula:
y - (-2) = -7/4(x - 4)
y + 2 = -7/4(x - 4)
y + 2 = -7/4x + 28/4
Rearrange the equation in slope-intercept form:
y = -7/4x + 28/4 - 2
y = -7/4x + 28/4 - 8/4
y = -7/4x + 20/4
y = -7/4x + 5
The equation of a line passing through the point (4, -2) and perpendicular to 4x+7y=8 is y = -7/4x + 5.
To find the slope-intercept equation for a line that is parallel to another line, we need to know that parallel lines have the same slope. So, we'll start by finding the slope of the given line.
The equation in standard form, 4x + 7y = 8, needs to be rearranged to slope-intercept form. The slope-intercept form is y = mx + b, where m represents the slope and b represents the y-intercept.
To convert the given equation to slope-intercept form, start by isolating y on one side:
4x + 7y = 8
7y = -4x + 8
y = (-4/7)x + (8/7)
Therefore, the slope of the given line is -4/7.
Since a line parallel to the given line has the same slope, we can use this information to build the equation for the parallel line. We will start with the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is the coordinate of the point the line passes through, and m is the slope.
Given the point (4,-2) and the slope -4/7, we'll substitute these values into the equation:
y - (-2) = (-4/7)(x - 4)
y + 2 = (-4/7)(x - 4)
Simplifying further will give us the slope-intercept form:
y + 2 = (-4/7)x + (16/7)
y = (-4/7)x + (16/7) - 2
y = (-4/7)x + (16/7) - (14/7)
y = (-4/7)x + (2/7)
Hence, the slope-intercept equation for a line passing through the point (4, -2) parallel to the line 4x + 7y = 8 is y = (-4/7)x + (2/7).
To find the slope-intercept equation for a line that is perpendicular to another line, we need to know that perpendicular lines have slopes that are negative reciprocals of each other. So, we'll start by finding the slope of the given line.
The equation 4x + 7y = 8 needs to be rearranged to slope-intercept form. Once again, the slope-intercept form is y = mx + b.
Begin by isolating y:
4x + 7y = 8
7y = -4x + 8
y = (-4/7)x + (8/7)
The slope of the given line is -4/7.
To find the slope of a line perpendicular to this line, we take the negative reciprocal of -4/7, which is 7/4.
Using the point-slope form as before, with the point (4, -2) and the slope 7/4:
y - (-2) = (7/4)(x - 4)
y + 2 = (7/4)(x - 4)
Simplifying further will give us the slope-intercept form:
y + 2 = (7/4)x - 7
y = (7/4)x - 7 - 2
y = (7/4)x - 9
Hence, the slope-intercept equation for a line passing through the point (4, -2) perpendicular to the line 4x + 7y = 8 is y = (7/4)x - 9.
7y = -4 x + 8
y = -(4/7) x + /7
so m = -4/7
y = -(4/7)x + b
-2 = -(4/7)(4) + b
-14/7 + 16/7 = b
b = 2/7
so
y = -(4/7)x + 2/7
7y = -4x + 2
for the second one
m = -(-7/4) = 7/4
so
y = (7/4) x + b
now put x = 4 and y = 2 in again to find b