Question 68·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The positive real numbers $A$ , $B$ , and $k$ satisfy

\frac{1}{A} + \frac{1}{B} = \frac{1}{k}.

Which equation correctly expresses $B$ in terms of $A$ and $k$ ?

The phrase express $B$ in terms of $A$ and $k$ signals rearranging a formula: keep $A$ and $k$ as letters while getting $B$ alone. Positivity lets you choose a valid numerical check in Desmos. Then subtract the fractions by hand using a common denominator. The key trap is stopping when $1/B$ is alone, because the question asks for $B$ .

Hints

Hint 1
Since $B$ is positive, $1/B$ adds a positive amount to $1/A$ . So $1/k$ is larger than $1/A$ . For positive numbers, which denominator must be smaller: $k$ or $A$ ?
Hint 2
Pick positive values with $k<A$ . In Desmos, use $B_1$ for the unknown and $\sim$ to find a value that fits the given equation. A numerical check can catch a sign error, but it cannot prove the formula.
Hint 3
A common denominator is one bottom both fractions can share. Use $kA$ to subtract $1/k-1/A$ . What you get is $1/B$ , not $B$ ; a reciprocal means $1$ divided by a number.

Step-by-step

Check a valid case, then isolate $B$
Step 1
Find the required order of $k$ and $A$
Because $1/B$ is positive, the equation says $1/k$ is larger than $1/A$ . For positive numbers with numerator $1$ , the larger fraction has the smaller denominator, so $k<A$ .
Step 2
Check the equation with allowed values
Choose $A=6$ and $k=2$ : both are positive, and $k<A$ . Type the three lines shown. Replacing $B$ with $B_1$ names the unknown, and $\sim$ tells Desmos to fit it. Desmos reports $B_1=3$ under PARAMETERS. This is a sign check, not the general formula.
Step 3
Get the reciprocal of $B$ alone
Subtract $1/A$ from both sides to isolate $1/B$ , the reciprocal of $B$ :
$\frac{1}{B}=\frac{1}{k}-\frac{1}{A}.$
Step 4
Give the fractions a common denominator
Use the common denominator $kA$ . Multiply each fraction's top and bottom by the factor its bottom is missing:
$\frac{1}{B}=\frac{A}{kA}-\frac{k}{kA}.$
Step 5
Subtract the numerators
The bottoms match now, so subtract only the tops:
$\frac{1}{B}=\frac{A-k}{kA}.$
Step 6
Turn the reciprocal into $B$
Since $k<A$ , $A-k$ is positive, so you can take the reciprocal of both sides: $B=\frac{kA}{A-k}$ .Getting $1/B$ alone is not getting $B$ alone. This expresses $B$ in terms of $A$ and $k$ . Choice C.

Question 68·Medium·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