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

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SAT Nonlinear Equations & Systems Question 133 | Free SAT Prep | Aniko

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Question 133·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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$3u + 2v = w - 7$

The given equation relates the variables $u$ , $v$ , and $w$ . Which equation correctly expresses $v$ in terms of $u$ and $w$ ?

When a question asks you to "express" one variable in terms of others, treat it as a straightforward solve-for-that-variable task: use inverse operations to isolate the target variable step by step. First, move any added or subtracted terms away from the target variable (carefully changing their signs when they cross the equals sign), then undo any multiplication or division by the coefficient on that variable. Finally, compare your simplified expression with the answer choices, paying close attention to signs and whether you have divided or multiplied by the coefficient.

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Focus on the goal

You want an equation that has $v$ alone on one side. Ask yourself: what steps will move all the other terms away from $v$ ?

First undo the addition

In $3u + 2v = w - 7$ , which term is attached to $v$ on the left side, and what operation is connecting it to $2v$ ? What can you do to both sides to remove the $3u$ from the left?

Then undo the multiplication

After you remove $3u$ from the left side, you will have an equation that starts with $2v = \,\dots$ . What operation will you use on both sides to turn $2v$ into just $v$ ?

Desmos Guide

Assign test values to $u$ and $w$

In Desmos, pick simple numbers for $u$ and $w$ (for example, type u = 1 and on the next line w = 10). You can later change these to other values to double-check.

Test each answer choice for $v$

For each option, define a corresponding $v$ in Desmos:

For choice A, type vA = 2*(w - 7) - 3*u
For choice B, type vB = (w - 7 - 3*u)/2
For choice C, type vC = (3*u + w - 7)/2
For choice D, type vD = (7 + 3*u - w)/2

Check which expression satisfies the original equation

Now compute the left side of the original equation for each $v$ and compare it to the right side w - 7:

Type LHS_A = 3*u + 2*vA
Type LHS_B = 3*u + 2*vB
Type LHS_C = 3*u + 2*vC
Type LHS_D = 3*u + 2*vD
Also type RHS = w - 7

See which $\text{LHS}$ value exactly equals RHS. That corresponding choice is the correct one. Try a second set of values for $u$ and $w$ to confirm your result.

Step-by-step Explanation

Understand what the question is asking

"Express $v$ in terms of $u$ and $w$ " means: rewrite the equation so that $v$ is alone on one side, and the other side has only $u$ , $w$ , and numbers.

We start with:

3u + 2v = w - 7

Our goal is to solve this equation for $v$ .

Isolate the $v$ term

To get the $v$ term by itself, remove $3u$ from the left side by subtracting $3u$ from both sides:

3u + 2v = w - 7 \\[0.7em] 3u + 2v - 3u = w - 7 - 3u

This simplifies to:

2v = w - 7 - 3u

Now $2v$ is isolated on the left.

Solve for $v$

Now we need to get $v$ alone. Since $2v$ means $2$ times $v$ , divide both sides of the equation by $2$ :

\frac{2v}{2} = \frac{w - 7 - 3u}{2}

This simplifies to:

v = \frac{w - 7 - 3u}{2}

So the equation that correctly expresses $v$ in terms of $u$ and $w$ is $v = \frac{w - 7 - 3u}{2}$ , which corresponds to choice B.

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Step-by-step Explanation

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