The equation relates the positive numbers and . Which equation correctly expresses in terms of ?

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SAT Nonlinear Equations & Systems Question 58 | Free SAT Prep | Aniko

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Question 58·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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The equation $\left(\dfrac{p}{3}\right)^{2} = r$ relates the positive numbers $p$ and $r$ . Which equation correctly expresses $p$ in terms of $r$ ?

For equations where a variable is inside a square, first rewrite so you have something like $p^2 = \text{(expression in } r\text{)}$ . Then take the square root of both sides, remembering to consider $\pm$ but using the given information (like the variable being positive) to choose the correct sign. Finally, simplify any radicals carefully—for example, use $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ so you do not confuse $3\sqrt{r}$ with $\dfrac{\sqrt{r}}{3}$ —and match your simplified expression to the answer choices.

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Get $p^2$ by itself

Try to rewrite $\left(\dfrac{p}{3}\right)^2$ in a simpler form and then isolate $p^2$ on one side of the equation.

Undo the square

Once you have an equation of the form $p^2 = \text{(something with } r\text{)}$ , what operation will you use to solve for $p$ ?

Use the positivity of $p$ and simplify the radical

After taking the square root, remember that $p$ is positive, and use $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ to simplify an expression like $\sqrt{9r}$ .

Desmos Guide

Pick a sample positive value for r

In Desmos, type something like r = 4 to set a specific positive value for $r$ (you can choose a different positive number if you prefer).

Define p for each answer choice

On separate lines, define

pA = sqrt(r)/3
pB = 3*r^2
pC = 9*r
pD = 3*sqrt(r) These correspond to the four options for $p$ in terms of $r$ .

Check which expression satisfies the original equation

On new lines, compute (pA/3)^2, (pB/3)^2, (pC/3)^2, and (pD/3)^2. Compare each result to the value of r. The correct expression for $p$ is the one whose (p/3)^2 equals r.

Verify with another value of r

Change the definition of r (for example, from r = 4 to r = 9) and watch how (pA/3)^2, (pB/3)^2, (pC/3)^2, and (pD/3)^2 change. The correct formula for $p$ will make (p/3)^2 match r again for this new value as well.

Step-by-step Explanation

Rewrite the equation to isolate $p^2$

Start with

\left(\frac{p}{3}\right)^2 = r.

Rewrite the left side as a single fraction squared:

\left(\frac{p}{3}\right)^2 = \frac{p^2}{9}.

So the equation becomes

\frac{p^2}{9} = r.

Now multiply both sides by $9$ to isolate $p^2$ :

9 \cdot \frac{p^2}{9} = 9r \quad\Rightarrow\quad p^2 = 9r.

Undo the square with a square root

To solve $p^2 = 9r$ for $p$ , take the square root of both sides:

p^2 = 9r \quad\Rightarrow\quad p = \pm \sqrt{9r}.

The $\pm$ appears because both a positive and a negative number square to give the same positive result.

Use the fact that $p$ is positive and simplify the root

We are told that $p$ is positive, so we keep only the positive root:

p = \sqrt{9r}.

Now simplify the square root using $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ :

\sqrt{9r} = \sqrt{9}\,\sqrt{r} = 3\sqrt{r}.

Therefore, the equation that correctly expresses $p$ in terms of $r$ is

p = 3\sqrt{r}.

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Step-by-step Explanation

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Step-by-step Explanation