Sure, let's solve the given equation [tex]\( y = \frac{\sqrt{t^2} - x^2}{v^3} \)[/tex] for [tex]\( x \)[/tex]:
1. Start with the given equation:
[tex]\[
y = \frac{\sqrt{t^2} - x^2}{v^3}
\][/tex]
2. Isolate the fraction by multiplying both sides by [tex]\( v^3 \)[/tex]:
[tex]\[
y \cdot v^3 = \sqrt{t^2} - x^2
\][/tex]
3. Simplify [tex]\( \sqrt{t^2} \)[/tex]:
[tex]\[
\sqrt{t^2} = |t|
\][/tex]
Therefore, the equation becomes:
[tex]\[
y \cdot v^3 = |t| - x^2
\][/tex]
4. Isolate the term involving [tex]\( x^2 \)[/tex] by subtracting [tex]\( |t| \)[/tex] from both sides:
[tex]\[
y \cdot v^3 - |t| = -x^2
\][/tex]
5. Change the sides of the equation for clarity:
[tex]\[
-x^2 = y \cdot v^3 - |t|
\][/tex]
Multiply both sides by -1:
[tex]\[
x^2 = - (y \cdot v^3 - |t|)
\][/tex]
Simplify the right-hand side:
[tex]\[
x^2 = -y \cdot v^3 + |t|
\][/tex]
6. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \pm\sqrt{-y \cdot v^3 + |t|}
\][/tex]
Thus, the solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = \sqrt{-y \cdot v^3 + |t|} \quad \text{and} \quad x = -\sqrt{-y \cdot v^3 + |t|}
\][/tex]
