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Problems:
1 Solve algebraically for a and b : 2 a – 3 b = 0 and a = 6.
2 Where do the lines y = – x + 5 and y = –1 cross? Find the answer algebraically.
3 Does the point (3 ; 4) lie on both line y = 4 and line y = – x + 1? Do algebraically.
4 Do the lines y = –2 and y = 2 intersect? Find the answer algebraically.
ACTIVITY 6
To solve simple exponential equations
[LO 2.4, 2.8]
Problems and some answers
1 I am thinking of a number that gives 100 when squared. What is the number?
The number could be 10, because 10 2 = 100. But is –10 not also a correct answer?
Yes, this problem has two valid answers!
Making an equation from this statement means starting by letting x be the number.
x 2 = 100
x 2 = 10 2 of x 2 = (–10) 2 The brackets are essential – can you see that?
x = 10 of x = –10 Both answers are valid.
2 I am thinking of a negative number that gives 25 when squared. What is it?
Let the number be y
y 2 = 25
y 2 = (5) 2 of y 2 = (–5) 2
y = 5 of y = –5 are the two solutions given by the equation.
According to the problem statement, though, only y = –5 is a valid answer.
3 There is a number that gives 27 when it is cubed. Find the number.
Let the number be x
x 3 = 27 x 3 = 3 3 x = 3.
Why can’t x be –3?
4 If I cube a certain number I get –8. What is the number?
5 Solve for x , and check your answers by the LHS/RHS method:
a) x 2 = 64 b) x 2 = 36 c) x 2 = –100 d) x 2 – 49 = 0
e) x 2 = 12,25 f) 3 x 2 = 12 g) 2 x 2 – 10,58 = 0
6 Solve for a and check your answers:
a) a 3 = 64 b) a 3 + 1 = 0 c) 2 a 2 = 16 d) a 4 = 81
Score yourself on the last 12 problems:
| LO 2 |
| Patterns, Functions and AlgebraThe learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills. |
| We know this when the learner: |
| 2.1 investigates, in different ways, a variety of numeric and geometric patterns and relationships by representing and generalising them, and by explaining and justifying the rules that generate them (including patterns found in nature and cultural forms and patterns of the learner’s own creation; |
| 2.2 represents and uses relationships between variables in order to determine input and/or output values in a variety of ways using: |
| 2.2.1 verbal descriptions; |
| 2.2.2 flow diagrams; |
| 2.2.3 tables; |
| 2.2.4 formulae and equations; |
| 2.3 constructs mathematical models that represent, describe and provide solutions to problem situations, showing responsibility toward the environment and health of others (including problems within human rights, social, economic, cultural and environmental contexts); |
| 2.4 solves equations by inspection, trial-and-improvement or algebraic processes (additive and multiplicative inverses, and factorisation), checking the solution by substitution; |
| 2.5 draws graphs on the Cartesian plane for given equations (in two variables), or determines equations or formulae from given graphs using tables where necessary; |
| 2.6 determines, analyses and interprets the equivalence of different descriptions of the same relationship or rule presented: |
| 2.6.1 verbally; |
| 2.6.2 in flow diagrams; |
| 2.6.3 in tables; |
| 2.6.4 by equations or expressions; |
| 2.6.5 by graphs on the Cartesian plane in order to select the most useful representation for a given situation; |
| 2.8 uses the laws of exponents to simplify expressions and solve equations; |
| 2.9 uses factorisation to simplify algebraic expressions and solve equations. |
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