2010 Paper 2 Q3 - simultaneous exponential

Without using a calculator, solve, for x and y, the simultaneous equations

For questions like these, whenever you seem stumped, start with the most fundamental idea. In this case, I would look at the common base. For the first equation, the common base is 2; for the second equation, the common base is 3. Let us convert the above questions to their common base, then…

For the first equation, this converts to

Or,          5x + y = 0 ---- (1)

The next equation converts to

And this becomes

Multiplying by x, we get

x² – 12x – 3xy = 4 ----- (2)

From (1),             y = -5x   ---- (3)

Substituting (3) into (2), we get

x² – 12x – 3x(-5x) = 4

x² – 12x + 15x² – 4 = 0

16x² – 12x – 4 = 0

Dividing throughout by 4, we get

4x² – 3x – 1 = 0

(4x + 1)(x – 1) = 0

x = - ¼ or x = 1

When x = -¼, y = 5/4

When x = 5, y = 5.