The equation relates the variables and . Which equation correctly expresses in terms of ?

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SAT Nonlinear Equations & Systems Question 115 | Free SAT Prep | Aniko

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Question 115·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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x^2 + y = 9

The equation relates the variables $x$ and $y$ . Which equation correctly expresses $y$ in terms of $x$ ?

When asked to "express one variable in terms of another," focus on isolating that variable by performing inverse operations step by step, always doing the same thing to both sides of the equation. For linear or simple nonlinear equations like this, move all other terms away from the target variable by adding, subtracting, multiplying, or dividing as needed, watch the signs carefully, and then match your simplified result to the answer choices.

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Focus on isolating y

You want an equation that has $y$ alone on one side and everything else on the other side. Look at the original equation and think about what is currently being done to $y$ .

Undo the operation with x²

In $x^2 + y = 9$ , $x^2$ is being added to $y$ . What operation will undo this so that only $y$ is left on the left side?

Do the same thing to both sides

Whatever you do to one side of the equation, you must do to the other. After you perform that operation, rewrite the new equation and see which choice matches.

Desmos Guide

Graph the original equation

In Desmos, type x^2 + y = 9. Desmos will graph this as a curve (a parabola opened downward).

Graph each answer choice as a function of x

On separate lines, type each option exactly as written, such as y = 9 - x^2, y = x^2 - 9, etc. Each will show a different graph.

Compare the graphs

Look for which $y = \dots$ graph lies exactly on top of (coincides with) the graph of x^2 + y = 9 everywhere. The matching graph corresponds to the correct equation for $y$ in terms of $x$ .

Step-by-step Explanation

Understand what "express y in terms of x" means

"Express $y$ in terms of $x$ " means rewrite the equation so that $y$ is alone on one side, and the other side is an expression that only involves $x$ and numbers.

We start with:

x^2 + y = 9

Isolate y by undoing the addition

On the left side, $y$ has $x^2$ added to it. To get $y$ alone, undo this by subtracting $x^2$ from both sides of the equation.

Subtract $x^2$ from both sides:

x^2 + y - x^2 = 9 - x^2

On the left, $x^2 - x^2$ cancels out, leaving just $y$ .

Write the simplified equation and match the choice

After canceling $x^2$ on the left, we get:

y = 9 - x^2

So the equation that correctly expresses $y$ in terms of $x$ is $y = 9 - x^2$ , which matches answer choice A.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Step-by-step Explanation