The equation above relates the variables and . Which expression gives in terms of ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 19 | Free SAT Prep | Aniko

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Question 19·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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3\,(x - 2)^2 + y = 5

The equation above relates the variables $x$ and $y$ . Which expression gives $y$ in terms of $x$ ?

When a question asks for one variable "in terms of" another, treat it like solving a simple linear equation: identify the variable you need to isolate (here, $y$ ), then use inverse operations to move all other terms to the opposite side, doing the same operation to both sides. Keep the entire expression together when you move it (like $3(x - 2)^2$ ) and pay close attention to signs so you don’t accidentally add when you should subtract or flip a sign incorrectly.

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Focus on isolating y

You want the equation in the form $y = \text{(something with } x \text{)}$ . What term is currently being added to $y$ on the left side?

Undo the operation with y

Since $3(x - 2)^2$ is added to $y$ , what operation can you do to both sides of the equation to remove $3(x - 2)^2$ from the left side?

Be careful with signs

When you move $3(x - 2)^2$ to the other side, does it become a plus or a minus on the right side? Think about what subtracting that whole term from both sides looks like.

Desmos Guide

Enter the original equation

Type the original equation exactly as given: 3*(x - 2)^2 + y = 5. Desmos will graph this relation (a parabola).

Test each answer choice

For each answer option, enter a new line in Desmos with that choice (for example, y = 5 + 3*(x - 2)^2, then try the others one by one).

Compare the graphs

For the correct choice, its graph will exactly overlap the graph of the original equation for all visible $x$ -values. If the graphs do not perfectly match, that answer choice does not represent the same relationship.

Step-by-step Explanation

Understand the goal

You are given the equation

3(x - 2)^2 + y = 5

The phrase "gives $y$ in terms of $x$ " means you need to solve this equation for $y$ , so that the equation looks like $y = \text{(expression with only } x \text{)}$ .

Isolate the y-term

In the equation $3(x - 2)^2 + y = 5$ , the term $3(x - 2)^2$ is added to $y$ .

To get $y$ by itself on the left side, you need to undo this addition by subtracting $3(x - 2)^2$ from both sides of the equation.

After you subtract $3(x - 2)^2$ from both sides, the left side will just be $y$ , and the right side will be $5$ minus that same term.

Write the new equation and match the answer choice

Subtracting $3(x - 2)^2$ from both sides gives

3(x - 2)^2 + y - 3(x - 2)^2 = 5 - 3(x - 2)^2

The $3(x - 2)^2$ terms cancel on the left, leaving

y = 5 - 3(x - 2)^2

Now look at the answer choices and select the one that matches this equation exactly: $y = 5 - 3(x - 2)^2$ .

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