Questions: Does the equation (5 v^3-7 u^2=1) define (v) as a function of (u)?

Transcript text: Does the equatio $5 v^{3}-7 u^{2}=1$ define $v$ as a function of $u$ ?

Solution

Solution Steps

Step 1: Understand the problem

The equation given is $5v^{3} - 7u^{2} = 1$. We need to determine if this equation defines $v$ as a function of $u$. In other words, for each value of $u$, does there exist exactly one corresponding value of $v$?

Step 2: Solve for $v$ in terms of $u$

To determine if $v$ is a function of $u$, we can attempt to solve the equation for $v$:

\[ 5v^{3} - 7u^{2} = 1 \]

First, isolate the term containing $v$:

\[ 5v^{3} = 1 + 7u^{2} \]

Next, divide both sides by 5:

\[ v^{3} = \frac{1 + 7u^{2}}{5} \]

Finally, take the cube root of both sides:

\[ v = \sqrt[3]{\frac{1 + 7u^{2}}{5}} \]

Step 3: Analyze the solution

The expression $v = \sqrt[3]{\frac{1 + 7u^{2}}{5}}$ shows that for each value of $u$, there is exactly one corresponding value of $v$. This is because the cube root function is a one-to-one function, meaning it maps each input to exactly one output. Therefore, $v$ is indeed a function of $u$.

Final Answer

Yes, the equation defines $v$ as a function of $u$.

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