Solve the simultaneous equations giving your answers to 3 s.f if necessary

Question

In(x-y) - 0
2 In x = ln (y+4)

Answer 1

We have the following simultaneous equations:

1) In(x-y) = 0
2) 2 In x = ln (y+4)

To solve this system of equations, we'll start with equation 1 and solve for x-y.

1) In(x-y) = 0
We know that ln(1) = 0, so x-y = 1.

Now, let's substitute x-y = 1 into equation 2:

2 In x = ln (y+4)
2 ln x = ln (y+4)
ln x^2 = ln (y+4)

Since the natural logarithm is a one-to-one function, we can equate the arguments:

x^2 = y+4

Now, we have a system of equations:

x - y = 1
x^2 - y = 4

We can solve this system by substitution or elimination method.

Using substitution, we can solve equation 1 for x and substitute into equation 2:

x = y + 1

Substituting into equation 2:

(y + 1)^2 - y = 4
y^2 + 2y + 1 - y = 4
y^2 + y - 3 = 0

We can solve this quadratic equation for y by factoring or using the quadratic formula.

(y - 1)(y + 3) = 0

This gives us two possible values for y: y = 1 or y = -3.

Case 1: y = 1
Substituting y = 1 into x - y = 1:
x - 1 = 1
x = 2

Case 2: y = -3
Substituting y = -3 into x - y = 1:
x - (-3) = 1
x + 3 = 1
x = -2

The solutions to the system of equations are (x, y) = (2, 1) and (-2, -3) rounded to 3 significant figures if necessary.