Y U(7,6)
T(3,4)
X
0
(k, k+8)
The figure shows a straight line passing through the points T(3, 4) and U(7,6).
(a) Find the gradient of TU.
(b) Find the equation of the line TU.
(c) A point (k, k+ 8) lies on the line TU produced. Find the value of k.

a) 1/2

b) y = 0.5x + 2,5

c) k = -3

Step-by-step explanation:

Given are points T(3, 4) and U(7,6).

To find the gradient of TU, you look at the incline in value in the y-direction, divided by the incline in value in the x-diretion.

(Uy- Ty) / (Ux- Tx)

substitute the values given and you get:

(6-4) / (7-3)

2 / 4

This gives you the answer on question a) The gradient of TU = 1/2

Any line is of the form y = ax + b with a = the gradient found in answer a) an b is the the value on the y-coordinate on the y axis, where the line TU intersects with the y-axis.

The whole next part is my attempt to explain to you something about getting the answer for b. Sorry if it is so long, but please take the time to read the explanation, and I hope it will help you to understand what is going on.

The gradient of TU = 1/2 which means that if you start on any point on the line, and then you go 2 units to the right and 1 up you will end upon another point on the line. That is because the quotient is the same. 2/4 = 1/2 = 0,5/1

The last quotient, 0,5/1 , also has the resulting value of 0,5.

This means that if you start on any point on the line, and then you go 1 unit to the right and 0.5 up, you will be on another point on the line.

ATTENTION : This means also, that if you start on any point on the line TU, and then you go 1 unit to the LEFT and 0.5 DOWN, you will be on another point on the line TU. If you understand this, than you have learned something important!

We need to know where the line TU intersects with the y-axis, and now we can use this to find that value?

Since point T(3,4) lies 3 units to the right of the y-axis, we need to go 3 units to the LEFT. But remember the incline of 0.5/1 ... For every unit you go to the left you need to go 0,5 unit DOWN to remain on the line.

If you go 3vunits to the left, how many units do you need to go down?

3 * 0.5 units which is 1.5.

Point T has an y coordinate of 4 so ... what will the y coordinate be if you go 3 units to the left?

4 - (1.5) = 2.5

So now finally we can write down the answer to question b).

y = 0.5x + 2,5

c) (k, k+ 8) lies on the line TU. So substitute that in y = 0.5x + 2,5

y = 0.5 * ( x ) + 2,5

k+8 = 0.5 * ( k + 8 ) + 2,5

k+8 = 0.5*k + 4 + 2,5

Bring all k values to the left of the equality sign, and all number values to the right.

k - 0.5*k = 4 + 2,5 - 8

0.5*k = 6,5 - 8

0.5*k = -1,5

multiply left and right of the = sign by 2 and you get your answer on question c).

k = -3