Question 920021: I am currently learning about the Rectangle Coordinate system and I am now looking for the equations using the Two-point form and the Point-slope form. We are required to present only one standard equation for all of our final answers.
The given points are (6,-4) and (8,-2) and the slope that I solved was 2/2 or 1. What should I do to make y= 6x-40 (equation after solving for point-slope) into 2x-2y-10=0(equation after solving for two-point) or vice versa?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your points are (6,-4) and (8,-2)
the formula for slope is (y2-y1) / (x2-x) which becomes:
slope = (-2--4) / (8-6) which becomes (-2+4) / 2 which becomes 2/2 which becomes 1.
the slope intercept form of the equation is y = mx + b
you have m which is the slope, so your equation becomes y = x + b
you now have to solve for b which is the y-intercept.
your points are (6,-4) and (8,-2).
you can use either point to solve for b.
2 ways to solve for b.
one way is to replace x and y with the values of one of the coordinates and solve for b.
for example, y = x + b becomes -4 = 6 + b when you replace y with -4 and x with 6.
solve for b to get b = -4 - 6 = -10
y = mx + b becomes y = x - 10 which is the slope intercept form of the equation.
another way is to use the point slope form of the equation and use one of the points.
the point slope form of the equation is y - y1 = m(x - x1)
m = 1
y1 = -4
x1 = 6
point slope form of the equation is y + 4 = 1(x-6) which becomes:
y + 4 = x-6
subtract 4 from both sides of this equation to get:
y = x - 10
that's the slope intercept form of the equation where m = 1 and b = -10.
now you have the slope intercept form of the equation and you want to convert that to the standard form of the equation.
standard form of the equation is ax + by = c
start with y = x - 10
subtract x from both sides of this equation to get:
y - x = -10
put the x in front of the y and you get:
-x + y = -10
it's not quite in standard form because the coefficient of the x term should be positive.
multiply both sides of the equation by -1 to get:
x - y = 10
now it's in standard form.
the other thing you need to consider is that the coefficients of x and y should be integers, so if you have fractional coefficients, you need to multiply both sides of the equation by a multiple so that the denominators disappear.
for example:
if your equation was 1/3x + 2/4y = 6, then multiply both sides of the equation by a common multiple such as 12 and you will get 4x + 3y = 72
that doesn't apply to your problem since your coefficients are already integers.
your final equation is x - y = 10
to confirm this equation is correct, donvert it back into slope intercept form.
subtract x from both sides of the equation to get -y = -x + 10
multiply both sides of the equation by -1 to get y = x - 10.
that was the original slope intercept frorm of the equation that you started with so you're good.
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