O lvl A Maths: Simultaneous Equations

Solve the following simultaneous equations

(a)


(b) Given that y = ax^b + 10 and that y = 26 when x = 2 and y = 64
when x = 3, find the value of a and of b.

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Answer:

(a)


Substitute x = 3 + 2y into (2)
2(3 + 2y) + 3y = -1
6 + 4y + 3y = -1
7y = -7
y = -1

Substitute y = -1 into (1)
x = 3 + 2y
x = 3 + 2(-1)
x = 1



(b)
y = ax^b + 10

Substitute x = 2 and y = 26 into

y = ax^b + 10
26 = a(2)^b + 10
16 = a(2)^b ------------------ (1)

Substitute x = 3 and y = 64 into
y = ax^b + 10
64 = a(3)^b + 10
54 = a(3)^b ------------------ (2)

(2) / (1)



By comparison, b = 3

Substitute b = 3 into (1),
16 = a(2)³
16 = 8a
a = 2

O lvl A Maths: Simultaneous Equations

Solve the pair of simultaneous equations

5x + 2y = 6
2x - 3y = 10

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Answer:

5x + 2y = 6 ------- (1)
2x - 3y = 10 -------(2)

We can solve by 2 methods.

Method 1: By elimination

(1) × 3: 15x + 6y = 18
(2) × 2: 4z - 6y = 20
(3) + (4): 19x = 38
Thus, x = 2

Substitute x = 2 into (1)
10 + 2y = 6
y = -2

Hence, the solution is x =2 , y = -2


Method 2: By substitution

From (1),
x = (6 - 2y) / 5 ------ (5)

Substitute (5) into (2)
2 × (6 - 2y) / 5 - 3y = 10
2 (6 - 2y) - 15y = 50
12 - 4y - 15y = 50
-19y = 50 - 12 = 38
y = -2

Substitute y = -2 into (5)
x = (6 - 2(-2)) / 5
x = 2

Hence, the solution is x =2 , y = -2