write an equation in slope intercept form of the line passing through the given points and is perpendicular to the graph of the given equations
1) (2,-2) ; y=-x-2
2)(2,-1) ; y=-3/2x+6
3)(4,2) ; y=-3
4)(-2,3) ; y-1/2x1
if the given line has slope m, then your perpendicular line must have slope -1/m. Then use the point-slope form of the line you want.
So, for the first one, you need a line through (2,-2) with slope 1:
y+2 = 1(x-2)
Do the others in like wise...
1) To find the perpendicular line to y = -x - 2 passing through the point (2,-2), we first need to find the slope of the given equation.
The equation is in slope-intercept form, y = mx + b, where m represents the slope. In this case, the slope of the given line is -1.
For a line to be perpendicular to another line, the slopes must be negative reciprocals of each other. Therefore, the slope of the perpendicular line will be the negative reciprocal of -1, which is 1.
Now that we have the slope, m = 1, and a point, (2,-2), we can use the point-slope form of an equation to find the equation of the perpendicular line.
The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Plugging in the values, we get:
y - (-2) = 1(x - 2)
y + 2 = x - 2
To rewrite the equation in slope-intercept form, we isolate y:
y = x - 4
Therefore, the equation in slope-intercept form for the line perpendicular to y = -x - 2 passing through (2,-2) is y = x - 4.
2) To find the equation of the line perpendicular to y = -3/2x + 6 passing through the point (2,-1), we first need to find the slope.
The given line is already in slope-intercept form, y = mx + b. The slope is represented by -3/2.
Again, for a line to be perpendicular, the slopes must be negative reciprocals of each other. So, the slope of the perpendicular line will be the negative reciprocal of -3/2, which is 2/3.
Using the point-slope form of an equation, y - y1 = m(x - x1), we can write:
y - (-1) = 2/3(x - 2)
y + 1 = 2/3x - 4/3
To rewrite the equation in slope-intercept form, we isolate y:
y = 2/3x - 4/3 - 1
y = 2/3x - 4/3 - 3/3
y = 2/3x - 7/3
Therefore, the equation in slope-intercept form for the line perpendicular to y = -3/2x + 6 passing through (2,-1) is y = 2/3x - 7/3.
3) To find the equation of the line perpendicular to y = -3 passing through the point (4,2), we first need to find the slope.
The given line is a horizontal line with a slope of 0.
For a line to be perpendicular, the slope must be the negative reciprocal of 0, which is undefined. In this case, we have a vertical line.
The equation of a vertical line passing through (4,2) will be of the form x = a, where a is the x-coordinate of the point.
Therefore, the equation in slope-intercept form for the line perpendicular to y = -3 passing through (4,2) is x = 4.
4) To find the equation of the line perpendicular to y - 1/2x = 1 passing through the point (-2,3), we first need to rearrange the given equation into slope-intercept form (y = mx + b).
Starting with the original equation:
y - 1/2x = 1
Adding 1/2x to both sides:
y = 1/2x + 1
We can see that the slope of this line is 1/2.
For a line to be perpendicular, the slope must be the negative reciprocal of 1/2, which is -2.
Using the point-slope form of an equation, y - y1 = m(x - x1), we can write:
y - 3 = -2(x - (-2))
y - 3 = -2(x + 2)
y - 3 = -2x - 4
To rewrite the equation in slope-intercept form, we isolate y:
y = -2x - 4 + 3
y = -2x - 1
Therefore, the equation in slope-intercept form for the line perpendicular to y - 1/2x = 1 passing through (-2,3) is y = -2x - 1.
To find the equation in slope-intercept form of a line that is perpendicular to a given line and passes through a given point, we need to follow these steps:
1) Find the slope of the given line.
2) Determine the negative reciprocal of the slope found in step 1 to get the slope of the perpendicular line.
3) Use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the line found in step 2.
4) Rearrange the equation in point-slope form to slope-intercept form, y = mx + b, where b is the y-intercept.
Let's apply these steps to the given points and equations:
1) Point: (2, -2), Equation: y = -x - 2
Slope of the given line: -1 (from the coefficient of x)
Negative reciprocal of -1: 1 (since the negative reciprocal of -1 is 1)
Using point-slope form: y - (-2) = 1(x - 2)
y + 2 = x - 2
Rearranging to slope-intercept form: y = x - 4
2) Point: (2, -1), Equation: y = -3/2x + 6
Slope of the given line: -3/2 (from the coefficient of x)
Negative reciprocal of -3/2: 2/3 (since the negative reciprocal of -3/2 is 2/3)
Using point-slope form: y - (-1) = 2/3(x - 2)
y + 1 = 2/3x - 4/3
Rearranging to slope-intercept form: y = 2/3x - 7/3
3) Point: (4, 2), Equation: y = -3
The given line is a horizontal line, and its slope is 0 since it doesn't involve x.
Negative reciprocal of 0 is undefined since there is no reciprocal for 0.
For a perpendicular line to a horizontal line with slope 0, the slope will be undefined.
Using point-slope form: (x - 4) has no coefficient since the slope is undefined
y - 2 = undefined * (x - 4)
y - 2 = undefined
The equation in slope-intercept form remains as: y = -3
4) Point: (-2, 3), Equation: y - 1/2x = 1
To put the given equation in slope-intercept form, we need to solve for y.
y - 1/2x = 1
y = 1/2x + 1
Slope of the given line: 1/2 (from the coefficient of x)
Negative reciprocal of 1/2: -2 (since the negative reciprocal of 1/2 is -2)
Using point-slope form: y - 3 = -2(x - (-2))
y - 3 = -2(x + 2)
y - 3 = -2x - 4
Rearranging to slope-intercept form: y = -2x - 1