SIMULTANEOUS EQUATIONS-1

Solve the following equations by substitution method.

2x - y = 5

x + 3y = 27

Solution

2x - y = 5 ----------- (i)

x + 3y = 27 ---------- (ii)

From equation (i)

2x - y = 5

2x - 5 = y

∴ y = 2x - 5 ----------(iii)

substitute equation (iii) in (ii) above,

x + 3(2x – 5 ) = 27

x + 6x – 15  = 27

x  + 6x  = 27 + 15

 7x  = 42                   [since x+6x=7x and 27+15=42]

7x  = 42

                  

7x = 42

7      7

x = 6

From equation (iii)

y = 2x - 5 ----------(iii)

y = 2(6) – 5

y = 12 – 5

y = 7

Hence x=6 and y=7.

TRY THIS………………………..

Solve the following equations by substitution method.

x + y = 18

2x - y = 14

ALGEBRA-2

Divide 68ax + 34ay - 51az by 17a.

solution

= 68ax + 34ay - 51az

              17a

= ⁴68ax  +  ²34ay  -  ³51az       [cancelling by 17a throughout]

     17a      17a          17a

= 4x + 2y - 3z.

TRY THIS...................

Divide 80am - 32an + 52ap by 2a

ARITHMETIC PROGRESSION-1

The 1st term of an A.P. is 73 and the common difference is 16. Find the 10th term.

Solution

A₁= 73, d = 20

A_n = A₁ + (n-1)d

A₁₀ = A₁ + (10-1)d

A₁₀ = A₁ + 9d

A₁₀ = 73 + (9x16) [ after substituting A₁= 73, d = 16 as given above]

∴ A₁₀ = 73 + 144

A₁₀ = 217

Hence the 10th term is 217.

TRY THIS……………

The 1st term of an A.P. is 76 and the common difference is 49. Find the 9th term.

APPROXIMATIONS-1

Estimate the value of 27 x 71.

solution

round off to tens

27 ≈ 30  [approximating to 30]

71 ≈ 70  [approximating to 70]

  = 30 x 70

  = 2100

Hence 27 x 71 ≈ 2100

TRY THIS...................

Estimate the value of 62 x 66. {ans. 4200}

INEQUALITIES-1

If 3x – 47 ≤ x + 5 ≤ 21x; find x

solution

3x – 47 ≤ x + 5 and  x + 5 ≤ 21x

3x – x ≤ 47+ 5 and  5 ≤ 21x - x

2x ≤ 52(divide by 2 both sides) and  5 ≤ 20x (divide by 20 both sides)

x ≤ 26 and  ⁵/20 ≤ x

x ≤ 26 and  ¹/4 ≤ x   [since  ⁵/20 simpifies to ¹/4 ]

x ≤ 26 and  x ≥ ¹/4 

TRY THIS........................

If 13x – 23 ≤ x + 25 ≤ 6x; find x.

DIFFERENCE OF 2 SQUARES-1

Evaluate 3994² – 3984²

Solution

3994² – 3984² = (3994 + 3984)( 3994 – 3984)

                     = (7978)( 10)     [or 7978 x 10]

                     = 79780

Hence 3994² – 3984² = 79780

TRY THIS…………….

Evaluate 631² – 531²

LOGARITHMS-1

Evaluate Log₂(256 ÷ 4).

Solution

= Log₂(256 ÷ 4)

= Log₂256 -  Log₂4          (applying the quotient rule)

= Log₂2⁸ -  Log₂2²             ( 256= 2⁸ and 4=2² )

= 8Log₂2 -  2Log₂2       ( remember  Log_aa^c = cLog_aa )

= (8 x 1) - (2 x 1)          ( remember  Log_aa = 1 )

= 8 - 2

= 6

hence Log₂(128 ÷ 4) = 6 

TRY THIS...............

Evaluate Log₂(1024 ÷ 64). 

APPROXIMATIONS-2

Estimate the value of 5.2  x  0.034

solution

= 5.2  x  0.034

= 5.0 x 0.03      [ 5.2 ≈ 5.0 to ones and 0.034 ≈ 0.03 to hundredths]

= 0.15

TRY THIS...................................

Estimate the value of 4.4  x  0.074

ALGEBRA-1

What is x if 2x + 5y = 40 and y=6?

solution

2x + 5y = 40

2x + 5(6) = 40;           [since y=6.]

2x + 30 = 40

2x = 40 - 30

2x = 10

2x = 10

2      2

x = 5

Hence x = 5

TRY THIS.........................

NECTA FORM II - 2013 QN. 9

What is x if x + 2y = 16 and y=6?

POLYGONS-1

A regular polygon has 55 sides. Find the total angles of that polygon.

Solution

n = 88

Total angles = (n - 2)180⁰

                   = (55 – 2)180⁰

                   = 53 x 180⁰

                   = 9540⁰

Total angles = 9540⁰

TRY THIS……………………….

A regular polygon has 34 sides. Find the total angles of that polygon.