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5.4 Equation of a Straight Line

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5.4 Equation of a Straight Line: y = mx + c

1. Given the value of the gradient, m, and the y-intercept, c, an equation of a straight line
y = mx + c can be formed.

2. If the equation of a straight line is written in the form y = mx + c, the gradient, m, and the y-intercept, c, can be determined directly from the equation.

Example:
Given that the equation of a straight line is y = 3 – 4x. Find the gradient and y-intercept of the line?
 
Solution:
y= 3 – 4x
y= – 4x + 3 ← (y = mx + c)
Therefore, gradient, m = – 4
y-intercept, c = 3



3. If the equation of a straight line is written in the form ax + by + c = 0, change it to the form y = mx + c before finding the gradient and the y-intercept.

Example:
Given that the equation of a straight line is 4x + 6y– 3 = 0. What is the gradient and y-intercept of the line?

Solution:
4x + 6y – 3 = 0
6y = –4x + 3

y=− 2 3 x+ 1 2 ← y=mx+c ∴ Gradient m=− 2 3      y−intercept, c= 1 2
 

5.5 Parallel Lines (Sample Questions)

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Example 1:

















The straight lines MN and PQ in the diagram above are parallel. Find the value of q.

Solution:
If two lines are parallel, their gradients are equal.
m1 = m2
mMN = mPQ

using gradient formula y 2 − y 1 x 2 − x 1 9 − 4 5 − ( − 1 ) = q − ( − 5 ) 5 − ( − 7 ) 5 6 = q + 5 12
60 = 6q + 30
6q = 30
q = 5

5.5 Parallel Lines (Part 1)

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5.5 Parallel Lines (Part 1)
 
(A) Gradient of parallel lines

1. Two straight lines are 
parallel if they have
the same gradient.
If PQ // RS,
then mPQ = mRS
   
2. If two straight lines have 
the same gradient, then  
they are parallel. 
If mAB = mCD
then AB // CD


Example 1:
Determine whether the two straight lines are parallel.
(a) 2y – 4x = 6
  y = 2x – 5
(b) 2y = 3x –4
  3y = 2x +12
 
Solution:
(a) 
2y – 4x = 6
2y = 6 + 4x
y = 2x + 3,   m1= 2
y = 2x – 5,   m2 = 2
m1= m2
Therefore, the two straight lines are parallel.
 
(b)
2y=3x−4 y= 3 2 x−2,    m 1 = 3 2 3y=2x+12 y= 2 3 x+4,    m 2 = 2 3 m 1 ≠ m 2 ∴ The two straight lines are not parallel.

5.6 SPM Practice (Short Questions)

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5.6.2 The Straight Line, SPM Paper 1 (Short Questions)

Question 6:
Diagram below shows a straight line RS with equation 3y = –px – 12, where p is a constant.
 

It is given that OR: OS = 3 : 2.
Find the value of p.

Solution:
Method 1:
Substitute x= –6 and y = 0 into 3y = –px – 12:
3(0) = –p (–6) – 12
0 = 6p – 12
–6p = –12
p = 2

Method 2:
OR: OS = 3 : 2
O R O S = 3 2 6 O S = 3 2 O S = 6 × 2 3 = 4 units  

Coordinates of S = (0, –4)
Gradient of the straight line RS = − − 4 − 6 = − 2 3  

Given 3y = –px – 12
Rearrange the equation in the form y = mx + c
y = − p 3 x − 4 Gradient of the straight line R S = − P 3 ∴ − P 3 = − 2 3 P = 2



Question 7:

 
The above diagram shows two straight lines, KL and LM, on a Cartesian plane. The distance KL is 10 units and the gradient of LM is 2. Find the x-intercept of LM.

Solution:


Let point N be = (0, 2).
Using Pythagoras’ Theorem,
 LN = √102 – 62 = 8
Point L = (0, 2 + 8) = (0, 10)
y-intercept of LM = 10
 
Using the gradient formula, m = − y-intercept x-intercept 2 = − ( 10 x-intercept ) x-intercept of L M = − 10 2 = − 5  

4.3 Operations on Statements (Sample Questions 1)

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Example 1:
Form a compound statement by combining two given statements using the word ‘and’.
(a) 3 × 12 = 36
 7 × 5 = 35

(b) 5 is a prime number.
 5 is an odd number.

(c) Rectangles have 4 sides.
 Rectangles have 4 vertices.

Solution:
(a) 3 × 12 = 36 and 7 × 5 = 35
(b) 5 is a prime number and an odd number.
(c) Rectangles have 4 sides and 4 vertices.



Example 2:
Form a compound statement by combining two given statements using the word ‘or’.
(a) 16 is a perfect square. 16 is an even number.
(b) 4 > 3.  -5 < -1

Solution:
(a) 16 is a perfect square or an even number.
(b) 4 > 3 or -5 < -1.

4.1 Statements (Sample Questions)

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Example 1:
Determine whether the following sentences are statements or not. Give a reason for your answer.
(a) 3 + 3 = 8
(b) 9 – 4 = 5
(c) A pentagon has 5 sides.
(d) 4 is a prime number.
(e) Is 40 divisible by 3?
(f) Find the perimeter of a square with each side of 4 cm.
(g) Help!

Answer:
(a) Statement; it is a false statement.
(b) Statement; it is a true statement.
(c) Statement; it is a true statement.
(d) Statement; it is a false statement.
(e) Not a statement; it is a question.
(f) Not a statement; it is an instruction.
(g) Not a statement; it is an exclamation.





Deduction and Induction

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4.6 Deduction and Induction

(A) Reasoning by Deduction and Induction
1. Reasoning by deduction is a process of making a conclusion for a specific case based on a given general statement.

2. Reasoning by induction is a process of making a generalization based on specific cases.

Maths Tip
1. General statement  →  Special conclusion  → Deduction
2. Specific cases  →  General conclusion  →  Induction
Example: 
Determine whether the following conclusion is made based on a deductive reasoning or inductive reasoning.

(a) Area of triangle = ½ × Base × Height
(i) 

Area of ∆ ABC
= ½ × 7cm × 5cm
= 17.5 cm2 
  
(ii)

Area of ∆ DEF
= ½ × 7cm × 4cm
= 14 cm2
  
(b)
1 = 7 (1)2 – 6
22 = 7 (2)2 – 6
57 = 7 (3)2 – 6
106 = 7 (4)2 – 6
 7n2 – 6, n = 1, 2, 3, 4…
 
Solution: 
(a)
The specific conclusion is made based on a general statement ~ Area of triangle = ½ × Base × Height. Therefore, the conclusion is made based on deductive reasoning.

(b) 
The general conclusion 7n2 – 6, n = 1, 2, 3, 4… is made based on specific cases. Therefore, the conclusion is based on inductive reasoning.

4.6 SPM Practice (Long Questions)

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Question 1:
(a) State if each of the following statements is true or false.
 (i) (i)  2 4 =16  and  12÷ 27 3 =3.
 (ii) 17 is a prime number or an even number.

(b) Complete the statement, in the answer space, to form a true statement by using the quantifier ‘all’ or ‘some’.

(c) Write down two implications based on the following statement:

A number is a prime number if and only if it is only divisible by 1 and itself.
Solution:
(a)(i) False
(a)(ii) True

(b)   Some   multiples of 3 are multiples of 6.

(c) Implication 1: If a number is a prime number, then it is only divisible by 1 and itself.
 Implication 2: If a number is only divisible by 1 and itself, then it is a prime
 number.


Question 2:
(a) State if each of the following statements is true or false.
  (i)  2 × 3 = 6 or 2 + 3 = 6
  (ii) 2 is a prime number and 5 is an even number.

(b) Write down the converse of the following implication.
  Hence, state whether the converse is true or false.

If x is a multiple of 12,
then x is a multiple of 3.
(c) Complete the premise in the following argument:

Premise 1: All hexagons have six sides
Premise 2: _____________________
Conclusion: ABCDEF has six sides.
Solution:
(a)(i) True
(a)(ii) False

(b) Converse: If x is a multiple of 3, then x is a multiple of 12.
  The converse is false.

(c) Premise 2: ABCDEF is a hexagon.

4.6 SPM Practice (Long Questions)

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Question 3:
(a) Complete each of the following statements with the quantifier ‘all’ or ‘some’ so that it will become a true statement.
  (i) ___________ of the prime numbers are odd numbers.
 (ii) ___________ pentagons have five sides.

(b) Write down two implications based on the following statement:

A ∩ B = B if and only if A υ B = A.
(c) Complete the premise in the following argument:

Premise 1: If a number is a factor of 24, then it is a factor of 48.
Premise 2: 12 is a factor of 24.
Conclusion: _____________________

Solution:
 (a)(i)    Some  of the prime numbers are odd numbers.
 (a)(ii)  All pentagons have five sides.

(b) Implication 1: If A ∩ B = B, then A υ B = A.
  Implication 2: If A υ B = A, then A ∩ B = B.

(c) Conclusion: 12 is a factor of 48.


Question 4:
(a) Combine the following two statements to form one true statement.
  Statement 1: (– 3)² = 9
  Statement 2: –3 (3) = 19
(b) Complete the premise in the following argument:

Premise 1: _____________________
Premise 2: x is a multiple of 25.
Conclusion: x is a divisible of 5.
(c) Make a general conclusion by induction for the sequence of numbers 7, 14, 27, … which follows the following pattern.
7 = 3 (2)1 + 1
14 = 3 (2)2 + 2
27 = 3 (2)3 + 3
…. = ………..
Solution:
(a) (– 3)² = 9 or –3 (3) = 19.

(b) Premise 1: All multiples of 25 is divisible by 5.

(c) 3 (2)n + n, where n = 1, 2, 3, …

4.2 Quantifiers ‘All’ and ‘Some’ (Sample Questions)

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Example 1:
Complete each of the following statements using the quantifiers ‘all’ or ‘some’ to make the statement true.
(a) _____ rectangles are squares.
(b) _____ prime numbers are odd numbers.
(c) _____ triangles have equal sides.
(d) _____ even numbers are divisible by 2.

Answer:
(a) Some rectangles are squares.
(b) Some prime numbers are odd numbers.
(c) Some triangles have equal sides.
(d) All even numbers are divisible by 2.



Example 2:
Construct a true statement using the quantifier ‘all’ or ‘some’ for the given object and property.
(a) Object: multiples of 4
 Property: can be divided exactly by 5
(b) Object: regular hexagon
 Property: 6 equal sides
(c) Object: acute angles
 Property: less than 90o

Answer:
(a) Some multiples of 4 can be divided exactly by 5.
(b) All regular hexagons have 6 equal sides.
(c) All acute angles are less than 90o.