Question 174·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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The equation

\left(\frac{k+3}{k-3}\right)^2=\frac{a}{b}

relates the positive real numbers $a$ , $b$ , and $k$ , with $k>3$ . Which equation correctly expresses $k$ in terms of $a$ and $b$ ?

A squared fraction in a nonlinear formula hides a sign choice: the condition on the requested variable tells you which square-root branch is allowed. For multiple choice, a fast route is to choose valid numbers, use a Desmos regression to find the permitted value, and test the proposed formulas. A formula that matches the squared equation can still fail the given restriction.

Hints

Hint 1
Because $k>3$ , both $k+3$ and $k-3$ are positive, and the numerator is larger. So their ratio is greater than $1$ . What simple positive values of $a$ and $b$ make $a/b$ greater than $1$ ?
Hint 2
A regression lets Desmos find a value that fits an equation. Replace $k$ with $k_1$ , change $=$ to $\sim$ , and restrict $k_1>3$ so Desmos finds the permitted branch.
Hint 3
An identity is a formula that works for every allowed input. Evaluate each proposed right side using your chosen $a$ and $b$ : one mismatch is enough to reject it. Which result matches the $k$ Desmos found?

Step-by-step

Approach 1: Test a valid case in Desmos
Step 1
Choose a valid test case
For $k>3$ , the fraction on the left is greater than $1$ , so choose positive values $a=4$ and $b=1$ .
Step 2
Find the permitted value of k
Type those values, then enter the given equation with $k_1$ in place of $k$ and $\sim$ in place of $=$ . The restriction $\{k_1>3\}$ tells Desmos to keep the permitted branch. Under PARAMETERS, Desmos shows $k_1=9$ .
Step 3
Shorten the formulas
Set $r=\sqrt{a/b}$ to shorten the choices; Desmos shows $r=2$ .
Step 4
Test the four formulas
Type their right sides in order, replacing each square root with $r$ . Desmos prints $9$ , $-1$ , $1$ , and $-9$ . Only the first result matches the permitted $k=9$ . One valid input disproves any proposed formula that gives the wrong value there. So the equation expressing $k$ is $k=\frac{3\left(1+\sqrt{\frac{a}{b}}\right)}{\sqrt{\frac{a}{b}}-1}$ . Choice A.

Approach 2: Rearrange the formula for every valid input
Step 1
Choose the square-root branch
Since $k>3$ , both $k+3$ and $k-3$ are positive, so their ratio is positive. Take square roots and keep the positive branch: $\frac{k+3}{k-3}=\sqrt{\frac{a}{b}}$ . Squaring hides a sign; the negative branch would contradict the sign of this fraction.
Step 2
Name the square root and its range
Write $r=\sqrt{a/b}$ . Because $k+3$ exceeds the positive denominator $k-3$ , their ratio is greater than $1$ , so $r>1$ . In particular, $r-1$ will not be zero when you divide by it.
Step 3
Clear the denominator
Multiply both sides by $k-3$ , which is nonzero because $k>3$ : $k+3=r(k-3)$ .
Step 4
Distribute r
Multiply $r$ by each term inside the parentheses: $k+3=rk-3r$ .
Step 5
Move the r term
Add $3r$ to both sides so the terms containing $r$ are no longer split across the equation: $k+3+3r=rk$ . Keep the whole $3r$ when you move it.
Step 6
Gather the k terms
Subtract $k$ from both sides to put the terms containing the requested variable together: $3+3r=rk-k$ .
Step 7
Factor both sides
A common factor is a multiplier shared by every term. Pull out $3$ on the left and $k$ on the right: $3(1+r)=k(r-1)$ . This leaves one factor of $k$ to isolate.
Step 8
Isolate k
Since $r>1$ , $r-1$ is positive. Divide both sides by it: $k=\frac{3(1+r)}{r-1}$ .
Step 9
Restore a and b
Replace $r$ with $\sqrt{a/b}$ to express $k$ in terms of the requested letters: $k=\frac{3\left(1+\sqrt{\frac{a}{b}}\right)}{\sqrt{\frac{a}{b}}-1}$ . The denominator is nonzero because $r>1$ . Choice A.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 174·Hard·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

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