How would I find point slope form, slope intercept form, and standard form for these three sets of coordinates. (5,3) (5,-1) (2,6) (4,-2) (-4,-3) (-1,-4)
Sorry, the coordinates were not in the correct order just ignore them and explain how to do point slope form, slope intercept form and standard form if you can please.
point slope relies on the fact that the slope m of a line is constant. So, for any point (x,y) on the line through (h,k), the slope
(y-k)/(h-k) = m
slope-intercept just reflects the fact that if the slope is m, every time x changes by 1, y changes by m.
so, y = mx
That intersects the axes at (0,0). If you know the y-intercept is y=b, the slope doesn't change, so
y = mx+b
don't know what you want to say about the standard form ax+by=c. That's just the way it's written.
To find the point-slope form, slope-intercept form, and standard form for a given set of coordinates, you need to calculate the slope of the line passing through the points and use it to determine the equations. Let's go through each set of coordinates and find the equations.
Set 1: (5,3), (5,-1), (2,6)
Step 1: Calculate the slope
We can find the slope between two points (x1, y1) and (x2, y2) using the formula:
slope = (y2 - y1) / (x2 - x1)
For the first pair of coordinates: (5, 3), (5, -1)
slope = (-1 - 3) / (5 - 5) = -4 / 0 (Undefined slope)
Since the slope is undefined, we have a vertical line passing through x = 5. Therefore, we cannot determine the point-slope and slope-intercept forms but can write the equation in standard form.
Step 2: Standard Form
The standard form equation of a line is Ax + By = C, where A, B, and C are constants. Since the vertical line passes through x = 5, the equation in standard form is:
x = 5
Set 2: (4,-2), (-4,-3), (-1,-4)
Step 1: Calculate the slope
For the second set of coordinates: (4, -2), (-4, -3)
slope = (-3 - (-2)) / (-4 - 4) = -1 / -8 = 1/8
Step 2: Point-Slope Form
The point-slope form of a line is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a given point. Let's use the first point (4, -2):
y - (-2) = (1/8)(x - 4)
y + 2 = (1/8)(x - 4)
Step 3: Slope-Intercept Form
The slope-intercept form of a line is y = mx + b, where m is the slope, and b is the y-intercept. Let's rearrange the point-slope equation to slope-intercept form:
y + 2 = (1/8)(x - 4)
y = (1/8)(x - 4) - 2
y = (1/8)x - 1/2 - 16/8
y = (1/8)x - 9/2
Step 4: Standard Form
The standard form equation of a line is Ax + By = C. Let's rearrange the slope-intercept form to standard form:
8y = x - 36
x - 8y = 36
Set 3: (-1,-4), (5,3), (2,6)
Step 1: Calculate the slope
For the third set of coordinates: (-1, -4), (5, 3)
slope = (3 - (-4)) / (5 - (-1)) = 7/6
Step 2: Point-Slope Form
Using the first point (-1, -4), we can write the point-slope form as:
y - (-4) = (7/6)(x - (-1))
y + 4 = (7/6)(x + 1)
Step 3: Slope-Intercept Form
Rearranging the point-slope equation to slope-intercept form:
y + 4 = (7/6)(x + 1)
y = (7/6)(x + 1) - 4
y = (7/6)x + 7/6 - 24/6
y = (7/6)x - 17/6
Step 4: Standard Form
Rearranging slope-intercept form to standard form:
6y = 7x - 17
7x - 6y = 17
In summary, the equations for the given sets of coordinates are:
Set 1:
Standard Form: x = 5
Set 2:
Point-Slope Form: y + 2 = (1/8)(x - 4)
Slope-Intercept Form: y = (1/8)x - 9/2
Standard Form: x - 8y = 36
Set 3:
Point-Slope Form: y + 4 = (7/6)(x + 1)
Slope-Intercept Form: y = (7/6)x - 17/6
Standard Form: 7x - 6y = 17