what is value of q? what id the value of y? how do you find the gradients of 4 lines?

Question

the line joining the point P(3,-4) to Q(q, 0) has a gradient of two. 

the three points X(2,-1), Y(8,y) and Z(11,2) are collinear....they lie on the same straight line.

Answer 1

Point P (3,-4) (know as Point 1, where x₁= 3 and y₁= -4), Point Q (q,0) (know as Point 2, where x₂= q and y₂₌ 0) , and the line containing these points has a slope (gradient) of 2, this is m (slope)

Use the slope formula to solve:

m = (y_{2 -} y₁) / (x₂ - x₁)

2 = (0- (-4)) / q - (3)

2 = 4 / q-3

2(q-3) =4

q-3 = 2

q = 5

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the three points X(2,-1), Y(8,y) and Z(11,2) are collinear....they lie on the same line.

If the three points are collinear, lying on the same line,

THEN THEY HAVE THE SAME SLOPE

Point 1 (2,-1), where x₁= 2 and y₁= -1, Point 2 (11,2) where x₂= 11 and y₂₌ 2)

to find the slope of the line

Use the slope formula to solve:

m = (y_{2 -} y₁) / (x₂ - x₁)

m = 2 -(-1) / 11 - (2)

m = 3/9 simplified m = 1/3

Now, use m = 1/3 to find the value of "y" :

Point 1 (2,-1), where x₁= 2 and y₁= -1, Point 2 (8,y) where x₂= 8 and y₂₌ y

Use the slope formula to solve:

m = (y_{2 -} y₁) / (x₂ - x₁)

1/3 = y - (-1) / 8 - 2

1/3 = (y+1) / 6

3(y+1) =6

divide both sides by 3

y+1 = 2

subtract 1 from both sides

y = 1