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Linear
equations
Clement
Radcliffe, Contributor
Let
us begin today's lesson by reviewing
the answers to last week's homework.
1.
Evaluate: (2r - 3)2
SOLUTION
(2r - 3)2 =
(2r - 3)(2r - 3) = 4r2 - 6r - 6r +
9
4r2
- 12r
+ 9
2.
Expand the following:
(a) (M+3)(M-4)
(b) (t-3)(t+6)
SOLUTION
(a)
(M+3)(M-4) = M2 + 3M -4M - 12 = M2
- M -12
(b) (t-3)(t+6) = t2 - 3t + 6t - 18
= t2 + 3t -18
3.
Evaluate:(-2p+1)(-3p - 6)
SOLUTION
(-2p+1)(-3p - 6) = 6p2 - 3p + 12p-6
= 6p2 + 9p -6
4.
Simplify (2y-1)/5 - (y+3)/2 The L.C.M.
of 5 and 2 is 10.
(
2(2y-1) -5(y+3) )/10
=
(4y-2-5y-15)/10 = (-y-17)/10
We
will now continue with LINEAR EQUATIONS.
LINEAR
EQUATIONS
The
inclusion of the EQUAL sign differentiates
from an algebraic expression. This
point is commonly missed by students
who sometimes attempt to solve algebraic
expressions. Do not fall into this
trap.
The
following points hsould be noted:
Equations identify either the relationship
between variables or the value of
a variable.
The
value of the variable is maintained
by performing identical operations
on both sides of the equation.
The
methods of clearing brackets and simplifying
algebraic expressions are usually
required to find solution of equations.
In
order to solve the equations one approach
is to simplify each side of the equation
and then equate both sides.
The
above is illustrated by the following
example:
Example
1
Solve
x + x = 6
3
x 4 = 6
x
= 24 = 8
x
= 3
Example
2
Solve
(4x + 5)/4 - (9+2x)/3 = 0
June
1996, No. 2 (d)
Considering
the left hand side, the L.C.M of 3
and 4 is 12
3(4x + 5) - 4(9 + 2x) = 12
12x
+ 15 - 36 -8x
=4x
- 21
=12
Equating
both sides:
(4x - 21)/12 = 0 (cross-multiplying)
4x - 21 = 0
x = 21/4
ALTERNATIVELY,
you may multiply all terms by the
L.C.M of the denominators.
(4x
+ 5)/4 - 9 + 2x/3 = 0
Multiply both sides by 12;
3(4x + 5) - 4(9 + 2x) = 0
12x + 15 - 36 - 8x = 0
4x - 21 = 0
x = 21/4
We
will now continue Algebra with the
topic Factorization.
Note that an algebraic expression
is factorized when it is expressed
as the product of its simplest factors.
The usual methods are:
(a)
Common factor
(b) Grouping
(c) Factorizing of Quadratic Expressions
(d) Difference of two squares.
The methods are adequately explained
in the text books and you should use
them to aid you as you revise for
your exams.
It
is important that you do the following
in all cases:
(a)
Bring each factor to its simplest
form,for example, a factor 16x + 8
should be expressed as 4(4x + 2)
(b) Check your answers, if you have
the time, by expanding and comparing
the result with the original expression.
Today
we will review the first two methods
of Factorization mentioned above.
EXAMPLES
OF COMMON FACTOR METHODS
1.
Factorize: 9x2 - 12x
The common factor method is used,
as 3x is the factor whic is common
to both terms.
Both terms are divided by 3x for us
to obtain the second factor.
Answer: 3x(3x-4)
2.
Factorize: 15x2y - 10xy3
Note that the common factor is 5xy
Answer is 5xy(3x - 2y2
EXAMPLES
OF GROUPING METHOD
3.
Factorize ax + ay + bx + by
Note that a is the common factor of
ax + ay and b the common factor of
bx + by
ax + ay + bx + by = a(x + y) + b(x
+ y)
Do
you realize that (x + y) is common
to both expressions?
a(x + y) + b(x + y) = (x + y)(a +
b)
This method could therefore be described
as repeated common factor method.
4.
Factorize 2ax - 6ay + bx - 3by
2a(x - 3y) + b(x - 3y)
= (x - 3y)(2a + b)
As
usual, I will close with close with
your Homework.
1.
Solve x/4 + 16 = 2x
2.
Solve (2x - 3)/2 - (x + 4)/4 = 1
3.
Factorize; (a) 7x2 - 21x
(b) axy - a2y
4.
Factorize: 3x - 8y - 4xy + 6
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A
Cumberland High School student
performs a poem in tribute to
National Hero, Norman Washington
Manley, during the Portmore
Municipal Council's Heroes Day
Civic Ceremony and Awards Presentation,
at the Portmore Pines Plaza,
recently. Some 15 persons, a
group and an organisation were
honoured by the council
- Anthony Minott/Freelance Photographer
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Clement
Radcliffe is the principal of Glenmuir
High School in May Pen.
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