ACTION Learning techniques for modelling linear problems
OBJECTIVE To learn some basic techniques to aid the modelling of linear problems
EXAMPLE 4 Find k ∈ R, if x = 1 is a solution of 2 (k x – 2) – 5 (2x – 1) = –3.
Solution Substituting x = 1 into 2 (k x – 2) – 5 (2x – 1) = –3 gives: 2 (k – 2) – 5 (1) = –3 2 k – 4 – 5 = –3 2 k = 6 k = 3
WORKED EXAMPLE
Mathematical modelling (word problems) Modelling a linear problem
A pipe 55 cm in length is cut into three pieces: a long piece, a short piece and an intermediate-length piece. The short piece is 20 cm shorter than the long piece. The long piece is twice as long as the intermediate piece. What are the lengths of all three pieces?
This is an example of a mathematical problem that is stated in words (a word problem). Mathematical modelling is the technique of translating these words into a mathematical equation and hence solving it.
1. Read the problem carefully and draw a diagram, if appropriate.
2. Assign a variable to one of the quantities to be found and hence assign variables to the other quantities. Let x = length of the intermediate piece
Long Intermediate Short 2xx (2x – 20)
3. Translate the word problem into an equation: 2x + x + (2x – 20) = 55 [All of the pieces add up to the total length.]
4. Solve the equation: 2x + x + 2x – 20 = 55 5x – 20 = 55 5x = 75 x = 15
