14 Algebra
Algebra gives us a distinctive way of solving problems, a deductive method with a twist.
That twist is ‘backwards thinking’. For a moment consider the problem of taking the
number 25, adding 17 to it, and getting 42. This is forwards thinking. We are given the
numbers and we just add them together. But instead suppose we were given the
answer 42, and asked a different question? We now want the number which when
added to 25 gives us 42. This is where backwards thinking comes in. We want the value
of x which solves the equation 25 + x = 42 and we subtract 25 from 42 to give it to us.
Word problems which are meant to be solved by algebra have been given to
schoolchildren for centuries:
My niece Michelle is 6 years of age, and I am 40.
When will I be three times as old as her?
We could find this by a trial and error method but algebra is more economical.
In x years from now Michelle will be 6 + x years and I will be 40 + x. I will be
three times older than her when
3 × (6 + x) = 40 + x
Multiply out the left-hand side of the equation and you’ll get 18 + 3x = 40 +
x, and by moving all the xs over to one side of the equation and the numbers to
the other, we find that 2x = 22 which means that x = 11. When I am 51 Michelle
will be 17 years old. Magic!
What if we wanted to know when I will be twice as old as her? We can use the
same approach, this time solving
2 × (6 + x) = 40 + x
to get x = 28. She will be 34 when I am 68. All the equations above are of the
simplest type – they are called ‘linear’ equations. They have no terms like x^2 or
x^3 , which make equations more difficult to solve. Equations with terms like x^2
are called ‘quadratic’ and those with terms like x^3 are called ‘cubic’ equations. In
past times, x^2 was represented as a square and because a square has four sides
the term quadratic was used; x^3 was represented by a cube.
Mathematics underwent a big change when it passed from the science of
arithmetic to the science of symbols or algebra. To progress from numbers to
letters is a mental jump but the effort is worthwhile.