Question 211·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Positive real numbers $q$ , $r$ , and $s$ satisfy

\frac{1}{q} + \frac{1}{r} = \frac{1}{s}

Which equation correctly expresses $s$ in terms of $q$ and $r$ ?

For a formula with several letters, substitution means replacing some letters with allowed numbers to test the answer choices. Pick values that keep every denominator nonzero, then use a Desmos regression on the given equation to find the requested value. A numerical match is a quick test, not a proof for every input. To prove the formula, combine the fractions and remember that their sum equals a reciprocal.

Hints

Hint 1
Choose positive, different values for $q$ and $r$ . Different values keep $q-r$ from being zero, so you can evaluate every answer choice with the same pair of numbers.
Hint 2
A regression asks Desmos to fit an unknown value. Replace the equation’s $=$ with $\sim$ and use $s_1$ as the unknown. Which answer choice gives that same value when you substitute your $q$ and $r$ ?

Step-by-step

Approach 1: Test the choices with Desmos
Step 1
Find a test value for $s$
Pick $q=2$ and $r=3$ . They’re positive and different, so every choice is defined. Type those values and $\frac{1}{q}+\frac{1}{r}\sim\frac{1}{s_1}\{s_1>0\}$ . The $\sim$ tells Desmos to fit $s_1$ , and the restriction keeps it positive as the problem requires. Under PARAMETERS, Desmos shows $s_1=1.2$ .
Step 2
Compare the choice values
Type the four right-hand sides using the same $q$ and $r$ . Desmos prints about $0.8333$ , $1.2$ , $-6$ , and $-0.1667$ , in that order. Only $\frac{qr}{q+r}$ matches the $s_1=1.2$ from the given equation. So $s=\frac{qr}{q+r}$ . A test pair identifies the choice; the algebra below shows why it works for every positive pair. Choice B.

Approach 2: Prove the formula with a common denominator
Step 1
Give the fractions one denominator
Since $q$ and $r$ are positive, $qr$ isn’t zero. Use $qr$ as a common denominator, multiplying each fraction’s top and bottom by the factor it’s missing:
$\frac{r}{qr}+\frac{q}{qr}=\frac{1}{s}.$
Step 2
Add the numerators
The denominators match, so add the tops and keep the plus sign:
$\frac{q+r}{qr}=\frac{1}{s}.$
This sum equals $1/s$ , not $s$ . Stopping here gives the tempting reversed fraction.
Step 3
Take the reciprocal to isolate $s$
A reciprocal flips a nonzero fraction. Both sides are positive, so take the reciprocal of each side:
$\frac{qr}{q+r}=s.$
So $s=\frac{qr}{q+r}$ for positive $q$ and $r$ . Choice B.

Question 211·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