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

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SAT Nonlinear Equations & Systems Question 89 | Free SAT Prep | Aniko

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Question 89·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The equation

3\sqrt{m} + 2n = p

relates the positive numbers $m$ , $n$ , and $p$ . Which equation correctly expresses $m$ in terms of $n$ and $p$ ?

For radical equations where you must "express one variable in terms of" others, treat the other letters as constants and use inverse operations step by step: first isolate the radical term, then undo multiplication or addition around it, and finally eliminate the radical by using the appropriate inverse (such as squaring for a square root). As you work, check that each algebraic step is applied to the entire expression (for example, square the whole fraction, not just its denominator) and match your final rearranged form exactly to the answer choices.

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Focus on isolating m

The phrase "express m in terms of n and p" means you should rewrite the equation so that m is alone on one side, with only n and p on the other side. Treat n and p like ordinary constants.

Move the n-term first

Look at the original equation $3\sqrt{m} + 2n = p$ . What operation will remove the $2n$ from the left side so that only the square root term involving m remains there?

Deal with the square root carefully

Once you have an equation of the form $\sqrt{m} = \text{(expression in n and p)}$ , what operation will turn $\sqrt{m}$ into m? Apply that operation to both sides and be careful to apply it to the entire expression on the right.

Desmos Guide

Represent the rearranged right-hand side

In a Desmos expression line, type p - 2n to represent the right-hand side after you conceptually subtract $2n$ from both sides of the original equation.

Divide by 3 to match the square-root side

In a new line, type (p - 2n)/3. This corresponds to the expression you get when you divide the previous result by 3, which matches the expression equal to $\sqrt{m}$ after isolating it.

Connect $\sqrt{m}$ to m and compare with choices

Think about how you would use this expression for $\sqrt{m}$ to write m itself (what operation undoes a square root?). Then, look through the answer choices and select the one whose right-hand side matches that resulting expression for m.

Step-by-step Explanation

Understand the goal and isolate the square root

We want an equation with m alone on one side and only n and p on the other.

Start with the given equation:

3\sqrt{m} + 2n = p

Subtract $2n$ from both sides to move the n-term away from the square root term:

3\sqrt{m} = p - 2n

Solve for the square root of m

Now isolate $\sqrt{m}$ by dividing both sides of the equation by 3:

\sqrt{m} = \frac{p - 2n}{3}

At this point, you have an expression for $\sqrt{m}$ in terms of n and p.

Undo the square root to solve for m

To get m (not its square root), apply the inverse operation of taking a square root, which is squaring.

Square both sides of the equation $\sqrt{m} = \frac{p - 2n}{3}$ :

(\sqrt{m})^2 = \left(\frac{p - 2n}{3}\right)^2

Since $(\sqrt{m})^2 = m$ , this simplifies to

m = \left(\frac{p - 2n}{3}\right)^2

So the correct equation expressing m in terms of n and p is $m = \left(\dfrac{p - 2n}{3}\right)^2$ . This matches choice B.

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Step-by-step Explanation

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