Higher Engineering Mathematics

ALGEBRA 3

A

Now try the following exercise.

Exercise 2 Further problems on brackets,

factorization and precedence

1. Simplify 2(p+ 3 q−r)−4(r−q+ 2 p)+p.
   [− 5 p+ 10 q− 6 r]


2. Expand and simplify (x+y)(x− 2 y).
   [x^2 −xy− 2 y^2 ]


3. Remove the brackets and simplify:
   24 p−[2{3(5p−q)−2(p+ 2 q)}+ 3 q].
   [11q− 2 p]


4. Factorize 21a^2 b^2 − 28 ab [7ab(3ab−4)].


5. Factorize 2xy^2 + 6 x^2 y+ 8 x^3 y.
   [2xy(y+ 3 x+ 4 x^2 )]


6. Simplify 2y+ 4 ÷ 6 y+ 3 ×[ 4 − 5 y.
   2
   3 y

− 3 y+ 12

]

7. Simplify 3÷y+ 2 ÷y−1.

[

5

y

− 1

]

8. Simplifya^2 − 3 ab× 2 a÷ 6 b+ab.[ab]

1.3 Revision of equations

(a) Simple equations

Problem 11. Solve 4− 3 x= 2 x− 11.

Since 4− 3 x= 2 x−11 then 4+ 11 = 2 x+ 3 x

i.e. 15= 5 xfrom which,x=

15

5

= 3

Problem 12. Solve

4(2a−3)−2(a−4)=3(a−3)− 1.

Removing the brackets gives:

8 a− 12 − 2 a+ 8 = 3 a− 9 − 1

Rearranging gives:

8 a− 2 a− 3 a=− 9 − 1 + 12 − 8

i.e. 3 a=− 6

and a=

− 6

3

=− 2

Problem 13. Solve

3

x− 2

=

4

3 x+ 4

.

By ‘cross-multiplying’: 3(3x+4)=4(x−2)

Removing brackets gives: 9 x+ 12 = 4 x− 8

Rearranging gives: 9 x− 4 x=− 8 − 12

i.e. 5 x=− 20

and x=

− 20

5

=− 4

Problem 14. Solve

(√

t+ 3

√

t

)

=2.

√

t

(√

t+ 3

√

t

)

= 2

√

t

i.e.

√

t+ 3 = 2

√

t

and 3 = 2

√

t−

√

t

i.e. 3 =

√

t

and 9 =t

(b) Transposition of formulae

Problem 15. Transpose the formula

v=u+

ft

m

to makefthe subject.

u+

ft

m

=vfrom which,

ft

m

=v−u

and m

(

ft

m

)

=m(v−u)

i.e. ft=m(v−u)

and f=

m

t

(v−u)

Problem 16. The impedance of an a.c. circuit

is given byZ=

√

R^2 +X^2. Make the reactance

Xthe subject.