Question

Given that x' = y(x - vt) and t' = y(t - vx/c?), derive the equations for x and t in terms of x' and t'. Show the detail and steps of solving this set of linear equations with two unknowns.

Answer 1

To derive the equations for x and t in terms of x' and t', we can start by rearranging the equations given:

x' = y(x - vt) --> Equation 1
t' = y(t - vx/c²) --> Equation 2

Let's solve for x in Equation 1:

x' = y(x - vt)
Divide both sides by y:
x'/y = x - vt
Add vt to both sides:
x'/y + vt = x

Now, let's solve for t in Equation 2:

t' = y(t - vx/c²)
Divide both sides by y:
t'/y = t - vx/c²
Add vx/c² to both sides:
t'/y + vx/c² = t

So, the equations for x and t in terms of x' and t' are:

x = x'/y + vt
t = t'/y + vx/c²

These equations allow us to express x and t in terms of x' and t' given the values of y, v, and c.