Question

Solve the simultaneous equations: ( 16ˣ = 32( 2ʸ ) ), ( log₂ left( 4 + 4y/x right) = 3 )

Answer 1

The simultaneous equations have no solution.

Step-by-step explanation:

To solve the simultaneous equations:

Equation 1: 16ˣ = 32(2ʸ)

Equation 2: log₂(4 + 4y/x) = 3

First, we can simplify Equation 1 by dividing both sides by 32:

16ˣ/32 = 32(2ʸ)/32

0.5ˣ = 2ʸ

Now, let's solve Equation 2 for y:

log₂(4 + 4y/x) = 3

2³ = 4 + 4y/x

8 = 4 + 4y/x

4y/x = 4

y/x = 1

y = x

Now that we have y in terms of x, we can substitute it back into Equation 1:

0.5ˣ = 2ʸ

0.5ˣ = 2ˣ

Since both sides have the same base, we can equate the exponents:

0.5ˣ = 2ˣ

0.5 = 2

This equation has no solution, so the two simultaneous equations have no solution as well.