The nonzero real numbers , , and satisfy Which equation correctly expresses in terms of and ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 146 | Free SAT Prep | Aniko

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Question 146·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The nonzero real numbers $y$ , $z$ , and $k$ satisfy

y = 5k^3 z^{-2}.

Which equation correctly expresses $k$ in terms of $y$ and $z$ ?

For equations where you need to solve for a variable in terms of others, first rewrite any negative exponents as fractions with positive exponents, then use basic algebra: clear denominators, isolate the term containing the variable (like $k^3$ ), and finally apply the inverse operation (such as taking a square root or cube root). Always work step by step and only match your final simplified expression to the choices at the end, which helps prevent errors from rushed mental manipulation.

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Handle the negative exponent

Focus on the expression $z^{-2}$ in the equation $y = 5k^3 z^{-2}$ . How can you rewrite $z^{-2}$ using a positive exponent?

Eliminate the denominator

Once you rewrite $z^{-2}$ , you'll see a fraction with $z^2$ in the denominator. What can you multiply both sides of the equation by to remove that denominator?

Isolate $k$ step by step

After you get an equation with $k^3$ alone on one side, what operation will undo a cube (power of 3) to solve for $k$ ?

Desmos Guide

Pick specific values for y and z

In Desmos, assign convenient nonzero values to $y$ and $z$ (e.g., type y = 3 and z = 2).

Graph the equation to find k

Type w = 5*x^3*z^(-2) (using x for k) and w = y. Click on the intersection; the x-coordinate is the value of $k$ for your chosen $y$ and $z$ .

Test each answer choice

For each option, compute its value with your $y$ and $z$ . For example, type (y*z^2/5)^(1/3). The option that matches your graphical solution is correct.

Step-by-step Explanation

Rewrite the negative exponent

Start with the given equation:

y = 5k^3 z^{-2}

Recall that $z^{-2} = \dfrac{1}{z^2}$ , so rewrite the equation as:

y = 5k^3 \cdot \frac{1}{z^2} = \frac{5k^3}{z^2}

Clear the fraction and isolate the term with $k$

From

y = \frac{5k^3}{z^2},

multiply both sides by $z^2$ to get rid of the denominator:

y z^2 = 5k^3

Now divide both sides by $5$ to isolate $k^3$ :

\frac{y z^2}{5} = k^3

Solve for $k$ by taking the cube root

To solve for $k$ , take the cube root of both sides of

\frac{y z^2}{5} = k^3.

This gives

k = \left(\frac{y z^2}{5}\right)^{1/3}

which matches one of the answer choices.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation