Question 88·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

0 / 233

The positive numbers $m$ , $n$ , and $k$ satisfy the equation

mn^2 = 8k

Which equation correctly expresses $n$ in terms of $m$ and $k$ ?

In a nonlinear formula, a squared letter must be isolated before you take its square root. The cue is a request to express one letter in terms of the others: keep those other letters fixed and undo multiplication first. A Desmos check with sample values can catch a slip, but it cannot prove a formula works for every value. Dividing twice does not undo a square.

Hints

Hint 1
Here, $m$ multiplies $n^2$ . Since $m$ is positive, you can divide both sides by $m$ . What expression does that leave for $n^2$ ?
Hint 2
A square root undoes squaring. Both a positive and a negative number can have the same square, so use the condition that $n$ is positive to choose the right sign.

Step-by-step

Isolate the square, then take its root
Step 1
Get $n^2$ alone
The equation has $m$ multiplying $n^2$ . Divide both sides by $m$ ; the problem says $m$ is positive, so it cannot be zero:
$n^2=\frac{8k}{m}.$
Step 2
Check one positive case in Desmos
For a sample check, type $m=2$ , $k=4$ , and $mn_{1}^{2}\sim8k\left\{n_{1}>0\right\}$ . The $\sim$ tells Desmos to find a value that fits the equation, and the braces restrict it to positive values. Under PARAMETERS, Desmos reports $n_{1}=4$ . That's one valid case, not a formula for every $m$ and $k$ .
Step 3
Undo the square and choose the sign
Dividing by $m$ isolated $n^2$ , not $n$ . Take the square root of both sides to undo the square. A negative number could have the same square, but $n$ must be positive, so use the positive square root:
$n=\sqrt{\frac{8k}{m}}.$
That expresses $n$ in terms of $m$ and $k$ . Choice C.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 88·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this