C11.

It may not be obvious at first, but this is actually just a 2-tier simultaneous equation, and whoever set it must’ve been really smart, because this question tests your ability to substitute numbers and play with fractions en route to solving the equation.

First thing we should do is to try and get rid of the fractions. Solving simultaneous equations are hard enough, we will try to minimise the usage of fractions. Hence,

Let a = 1 / (x + y) and let b = 1 / (x – y)

Immediately, the question becomes a lot more straightforward, and you’re solving two simultaneous equations involving a and b.

Rewriting the two equations:

5 a + b = 9

b = 9 – 5 a — (1)

2 a – 3 b = 7 — (2)

Sub (1) into (2):

2 a – 3 (9 – 5 a) = 7

2 a – 27 + 15 a = 7

17 a = 34

a = 2

Sub a = 2 into (1)

b = 9 – 5 (2)

b = -1

Hence, 1 / (x + y) = 2 and 1 / (x – y) = -1

Now, we can solve the simultaneous equations involving x and y.

Rewriting the equations:

x + y = 1/2 — (3)

x – y = -1

x = y – 1 — (4)

Sub (4) into (3):

(y – 1) + y = 1/2

2y – 1 = 1/2

2y = 3/2

y = 3/4

Sub y = 3/4 into (3):

x + 3/4 = 1/2

x = -1/4//

B10.

If the graphs meet at (2 , 3), it means that they meet when x = 2 and y = 3.

Hence, substitute in x = 2 and y = 3 into both equations:

2 ax + 3 by = 10 -> 4 a + 9 b = 10 — (1)

3 ax – 7 by = -54 -> 6 a – 21 b = -54 — (2)

Since 4 and 6 are relatively easy to find a LCM (12), take the 1st equation and multiply by 3, and take the second equation to multiply by 2:

(1) x 3:

12 a + 27 b = 30 — (3)

(2) x 2:

12 a – 42 b = -108 — (4)

Once we have equations (3) and (4), we can use the elimination method to eliminate the ‘a’ term by subtracting one from the other. To keep the numbers positive, I’ll subtract (4) from (3) but if you reverse it, it is fine too.

(3) – (4):

12 a + 27 b – (12 a – 42 b) = 30 – (-108)

If you’re starting out, its better to put these brackets so you don’t accidentally flip the signs wrongly.

27 b – (-42 b) = 138

69 b = 138

b = 2

Once you find b, use b = 2 to sub into any of the equations you already have to find a:

Sub b = 2 into (1):

4 a  + 9 (2) = 10

4 a = 10 – 18

4 a = -8

a = -2//