The equation above relates the distinct positive numbers and . Which equation correctly expresses in terms of ?

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SAT Nonlinear Equations & Systems Question 85 | Free SAT Prep | Aniko

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Question 85·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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(x + 5)(y - 2) = 18

The equation above relates the distinct positive numbers $x$ and $y$ . Which equation correctly expresses $y$ in terms of $x$ ?

For problems that say “express $y$ in terms of $x$ ” and start with a factored equation, focus on isolating the group containing $y$ step by step. First, undo multiplications by dividing both sides by the factor attached to the $(y - \text{something})$ term, then undo any addition or subtraction to solve for $y$ completely. Avoid common errors like dividing by the wrong factor or changing constants; keep track carefully of which factor is multiplied with $y$ in the original equation.

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Focus on the structure of the equation

You are given $(x + 5)(y - 2) = 18$ and asked to express $y$ in terms of $x$ . Which part of this equation actually contains $y$ ?

Undo the multiplication

Notice that $y - 2$ is multiplied by $x + 5$ . What operation will undo this multiplication so that you can get $y - 2$ by itself?

Finish isolating y

Once you have $y - 2$ on one side, what final step do you need to perform to get just $y$ ?

Desmos Guide

Graph the original relation implicitly

In Desmos, type the original equation exactly as given:

(x+5)(y-2)=18

Desmos will graph this as an implicit curve (a hyperbola).

Graph each answer choice as a separate function

For each choice, type its equation into Desmos:

For A, type $y = 2 - 18(x + 5)$ .
For B, type $y = 18/(x + 3)$ .
For C, type $y = (x + 5)/18 + 2$ .
For D, type $y = 2 + 18/(x + 5)$ . Each of these will appear as its own curve.

Compare the graphs

Look at the graphs and see which $y =$ curve lies exactly on top of the curve from $(x+5)(y-2)=18$ for all visible points. The equation whose graph perfectly matches the original relation is the correct choice.

Step-by-step Explanation

Identify what you need to solve for

The question asks for an equation that expresses $y$ in terms of $x$ . This means you should rearrange the given equation

(x + 5)(y - 2) = 18

so that $y$ is alone on one side of the equation.

Isolate the factor containing y

In $(x + 5)(y - 2) = 18$ , the expression $y - 2$ is multiplied by $x + 5$ .

To isolate $y - 2$ , divide both sides of the equation by $x + 5$ :

\frac{(x + 5)(y - 2)}{x + 5} = \frac{18}{x + 5}

The $x + 5$ terms cancel on the left, leaving

y - 2 = \frac{18}{x + 5}.

Solve for y completely

Now get $y$ by itself. Add 2 to both sides of

y - 2 = \frac{18}{x + 5}

which gives

y = 2 + \frac{18}{x + 5}.

This matches answer choice D.

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Step-by-step Explanation

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Step-by-step Explanation