1. ## Mostly linear problems i had trouble with

Okay, so i did all the questions in the chapter and these are the questions i had trouble with. If u don't might could u show ur working becos i know wot the answes r, i just don't know how to get 2 them. Thanks alot

1 .P and Q are the points of intersection of the line y/2 + x/3= 1 with the x and y axes repectively. The gradient of QR is 1/2, where R is the point with x-coordinate 2a,a>0.
a. Find the y-coordinate of R in terms of a
b.Find the value of a if the gradient of PR is -2

2. In the a rectangle (diagram below) ABCD, A and B are the points (4,2) and (2,8) respectively. Given the equation of AC is y=x-2, find:
a. The equation of BC
b. the coordinates of C

3. ABCD is a parallelogram, lettered anticlockwise, such that A and C are the points (-1,5) and (5,1) repectively.
a. Given that BD is parallel to the line whose equation is y+5x=2, find the equation of BD.

4. A car journey of 300km lasts 4h. Part of this journey is on a freeway at an average speed of 90km/h. The rest is on country roads at an average speed of 70km/h. Let T be the time(in hours) spent on the freeway.
a. In terms of T, state the number of hours travelling on country road.

2. 1 .P and Q are the points of intersection of the line y/2 + x/3= 1 with the x and y axes repectively. The gradient of QR is 1/2, where R is the point with x-coordinate 2a,a>0.
a. Find the y-coordinate of R in terms of a
b.Find the value of a if the gradient of PR is -2
Part a:
P(3,0), Q(0,2), R(2a,?)
$\displaystyle m_{QR}=\frac{1}{2}$
If we find the rule for the line connecting points Q and R, we can then substitute 2a into that equation to find y in terms of a.
y intercept is 2 and gradient is 1/2, equation between Q and R is $\displaystyle y=\frac{1}{2}x+2$

$\displaystyle y=\frac{1}{2}\times 2a + 2$
$\displaystyle y=a+2$
so y coordinate or R in terms of a is $\displaystyle a+2$

Part b:
Equation of line R and P has gradient of -2 and x intercept at (3,0)
Finding the equation between the 2 points..
$\displaystyle y-y_{1}=m(x-x_{1})$
$\displaystyle y-0=-2(x-3)$
$\displaystyle y=-2x+6$

notice R is the intersection between the 2 lines with equations:
$\displaystyle y=\frac{1}{2}x+2$ and $\displaystyle y=-2x+6$
Simply let y=y in those 2 equations to find x coordinate of the intersection which is 2a.
$\displaystyle \frac{1}{2}x+2=-2x+6$
$\displaystyle x+4=-4x+12$multiply everything by 2 (so x is no longer a fraction)
$\displaystyle 5x=8$get x by itself
$\displaystyle x=\frac{8}{5}$
we were told to find a: now $\displaystyle \frac{8}{5}=2a$
$\displaystyle a=\frac{4}{5}$

3. Originally Posted by chaneliman
2. In the a rectangle (diagram below) ABCD, A and B are the points (4,2) and (2,8) respectively. Given the equation of AC is y=x-2, find:
a. The equation of BC
b. the coordinates of C
Strictly speaking the points A and C are on the line y = x - 2. Either you copied it wrong or your book stated it incorrectly.

Anyway:
a) We know that ABCD is a rectangle, so we know that segment AB is perpendicular to BC. So work out the details of the following procedure:
* Find the equation of the line that contains the points A and B.

* Call the slope of this line m. Then we know the slope of the line containing the points B and C is $\displaystyle -\frac{1}{m}$ since the two segments are perpendicular.

* You now have the slope of the line containing the points B and C. So use the coordinates of point B to find the y-intercept of this line. This will give you the equation of the line containing B and C.

b) You have the line containing points B and C from part a). The line containing points A and C is y = x - 2. These lines intersect at point C. So solve the system of equations represented by the two lines.

-Dan

4. Originally Posted by chaneliman
3. ABCD is a parallelogram, lettered anticlockwise, such that A and C are the points (-1,5) and (5,1) repectively.
a. Given that BD is parallel to the line whose equation is y+5x=2, find the equation of BD.
The midpoint of AC is (2,3) and the equation of BD must be of form y+5x = c. We have to find 'c'. Now since (2,3) should lie on it, 3+10 = c and hence the equation of BD is y+5x = 13

5. Hello, chaneliman!

4. A car journey of 300 km lasts 4 hours.
Part of this journey is on a freeway at an average speed of 90 km/h.
The rest is on country roads at an average speed of 70 km/h.

Let $\displaystyle T$ be the time(in hours) spent on the freeway.

(a) In terms of $\displaystyle T$, state the number of hours travelling on country roads.
Part (a) is a simple designation . . . Where is your difficulty?

We are told: .$\displaystyle \text{Let }T = \text{hours on the freeway}$

Since the entire jouney took 4 hours,
. . then $\displaystyle 4-T$ hours were spent on country roads.

6. thanks for that guyz,

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# in a diagram of a rectangle A and B are the points (4,2) and (2,8) respectively. Given that the equation of AC is y=x-2, find the equation of BC

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