To eliminate the parameter \( t \) from the given parametric equations:
- The first equation is \( x(t) = 2t - 1 \).
- The second equation is \( y(t) = \sqrt{t} \).
First, we can solve the first equation for \( t \):
\[ x = 2t - 1 \implies 2t = x + 1 \implies t = \frac{x + 1}{2} \]
Next, we substitute this expression for \( t \) into the equation for \( y \):
\[ y = \sqrt{t} = \sqrt{\frac{x + 1}{2}} \]
This gives us:
\[ y = \sqrt{\frac{x + 1}{2}} = \frac{\sqrt{x + 1}}{\sqrt{2}} \]
However, we need to express the equation without the square root in the context of options provided in the responses.
Instead of working with the square root directly, we can square both sides of our equation for \( y \):
\[ y^2 = \left(\sqrt{t}\right)^2 = t \]
Substituting \( t \):
\[ y^2 = \frac{x + 1}{2} \]
Multiplying by 2 gives:
\[ 2y^2 = x + 1 \]
Now solving for \( x \):
\[ x = 2y^2 - 1 \]
Now, we can see from our derived equation that \( x \) is expressed in terms of \( y \):
\[ x = 2y^2 - 1 \]
Thus, the correct response based on the options provided is:
x equals 2 y squared minus 1, y is greater than or equal to 0.