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Re: GED

rdf8585 said:
"Write an equation of the line that passes through the point and had the specified slope. Sketch the line."

(0, 10), m = negative 1/4
m = rise/run
y = mx + b
10 = -(1/4)0 +b
10 = b

y = -(1/4)x + 10 -> if x = 0, y = 10; then plug in different X values to get the Y value.

"Use the point-slope form to write an equation of a line that passes through the oiunt and has the specified slope. Sketch the line."
(0, 2), m = 3/5
m = (y2-y1)/(x2-x1)
y = mx + b
(3/5) = (y-2)/(x) -> cross-multiply
3x = 5(y-2)
3x = 5y - 10
3x + 10 = 5y
(3/5)x + 2 = y -> y = mx + b; y = (3/5)x +2

"Write the point-slope form of the equation of the line."

(3, 1), m = 3/2
y = mx + b -> 3/2 is m, use this as a check
where m = (y2-y1)/(x2-x1)

let x1 = 3, y1 = 1 (starting point = x1, y1)
m = rise/run
m = (y2 - y1)/(x2 - x1)
(3/2) = (y - 1)/(x - 3) cross-multiply
3(x - 3) = 2(y - 1)
3x - 9 = 2y - 2
3x - 7 = 2y
(3/2)x - (7/2) = y -> y = mx + b -> y = (3/2)x + (7/2)

Check with the direct method: y = mx + b -> solve for b
1 = (3/2)3 + b
1 = (9/2) + b
(2/2) = (9/2) + b
-(7/2) = b
therefore y = (3/2)x - (7/2)

"Write an equation of the line that passes through the points. When possible, write the equation in slope-intercept form."

(6, -1), (3, 3)
First thing you are given is the two points -> solve for slope m
m = (y2-y1)/(x2-x1); let x1 = 6, y1 = -1, x2 = 3, y2 = 3 from the points given.
m = (3 - (-1))/(3 - 6)
m = (3 + 1)/(3 - 6)
m = (4/-3)
m = -(4/3) take either point, use y = mx + b to solve for b.
y = mx + b <- (3, 3) or (6, -1) -> y = mx + b
3 = -(4/3)3 + b -1 = -(4/3)6 + b
3 = -4 + b -(3/3) = (-24/3) + b
7 = b -1 = -8 + b ----> 7 = b!

Put the pieces together, leaving x and y as variables ("for any X and Y")
y = mx + b
y = -(4/3) + 7

"Write an equation of the line passing through the points"

(3, 5), (1, 6)
Same deal -> you are given two points, solve for the slope between them.
m = (y2-y1)/(x2-x1) -> let x1 = 3, y1 = 5, x2 = 1, y2 = 6
m = (6 - 5)/(1 - 3)
m = -(1/2) -> now plug either point into y= mx + b to solve for b.
y = mx + b <- (3, 5) (1, 6) -> y = mx + b
5 = -(1/2)3 + b 6 = -(1/2)1 + b
5 = -(3/2) + b 6 = -(1/2) + b
(10/2) = -(3/2) + b (12/2) = -(1/2) + b
(13/2) = b (13/2) = b

Plug back into y = mx +b for any x and y
y = -(1/2)x + (13/2)

Whew! Boy, that took a while to type out! I hope this makes sense. Fractions...well, I did the best I could with them! :lol
 
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