Linear equations
AS WE pursue exam-type materials, I expect that students are making greater effort to understand the materials provided and to continue to practise in earnest.
A BRIEF REVIEW OF ‘CHANGE THE SUBJECT OF THE FORMULA’ LINEAR EQUATIONS
The inclusion of the EQUAL sign differentiates an EQUATION from an algebraic expression. This point is commonly missed by students who sometimes attempt to solve algebraic expressions. Do not make this error.
The following points should 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 solutions of linear equations.
■ In order to solve linear equation, one approach is to simplify each side of the equation and then equate both sides.
EXAMPLE 1
Given that 2(x – 1) – 3x = 6, then x = a) – 8 b) – 4 c) 4 d) 8
Simplifying the left-hand side, 2(x – 1) – 3x = 2x – 2 – 3x = – x – 2 Equating
– x – 2 = 6 with – x = 6 + 2 = 8 ■ x = – 8, the answer is a)
EXAMPLE 2
Solve 3x + x = 14
2 4
Simplify the left-hand side:
3x + x = 6x + x = 7x 2 4 4 4 4
Equating both sides:
■ 7x = 14 Multiply both sides by 4 4
7x x 4 = 14 x 4 = 56
4
■ 7x = 56
■ x = 56 = 8 7 Ans: 8
EXAMPLE 3
Solve 4x + 5 – 9 + 2x = 0
4 3 Considering the left hand-side, the LCM of 3 and 4
is 12.
3(4x + 5) – 4(9 + 2x)
12
= 12x + 15 – 36 – 8x
12
= 4x – 21
12
Equating both sides:
■ 4x – 21 = 0 Multiply both sides by 12 12
■ 4x – 21 = 0 or 4x = 21
■ x = 21
4
We will now continue algebra with the topic Factorisation.
Note that an algebraic expression is factorised when it is expressed as the product of its simplest factors. For example, 6b + 15 is expressed as 3( 2b + 5)
The usual methods are:
(a) Common factor
(b) Grouping
(c) Factorising of quadratic expressions (d) Difference of two squares.
The methods are adequately explained in the textbooks 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 20x + 12 should be expressed as 4(5x + 3) or 2 – 6y as 2( 1 – 3y).
(b) Check your answers, if you have the time, by expanding the factors and comparing the result with the original expression.
This week, we will review the first two methods of factorisation mentioned above.
EXAMPLES OF COMMON FACTOR METHOD
1. Factorise 28xy + 7y 28xy + 7y = 7y ( 4x + 1) NB: 7y is the only factor common to both terms.
Checking, multiplying 7y ( 4x + 1) = 28xy + 7y, the given expression.
2. Factorise 16m + 4
The factors of 16m are 2, 4, 8 and m, and 4 has factors 2 and 4. The highest common factor (HCF) is 4.
16m + 4 = 4( 4m + 1 )
Answer: 4( 4m + 1 )
3. Factorise: 16x² – 12x
The common factor method is used, as 4x is the factor which is common to both terms. Both terms are divided by 4x for us to obtain the second factor.
Answer: 4x(4x – 3)
Please note that by expanding the answer, 4x(4x – 3) = 16x2 – 12x, the given expression
Please factorise each of the following on your own:
■ 4x2 y + 12xy
■ 8 – 12mn
■ 2b3 – 6 b2 – 4b
EXAMPLES OF GROUPING METHOD
5. Factorise 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 realise 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. Factorise 2ax – 6ay + bx – 3by
(2ax – 6ay) + (bx – 3by) Factorise each as follows:
2a(x – 3y) + b (x – 3y) = (x – 3y)(2a+ b)
As usual, I will close with your homework. 1. Solve for t,
MISCELLANEOUS EXAMPLE
9. PTRS is a quadrilateral. Q is a point on PT such that QT = QR = QP.
Angle QRT = 76o .
Determine, giving a reason for each step of your answer, the measure of: (i) Angle RQT
(ii) Angle PRT
(iii) Angle SPT, given that angle SRT = 145o and PSR = 100o.
Please identify additional questions to be included in your revision pool.