Isolate a hidden expression

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Turn bulky SAT systems into smaller equations by renaming repeated expressions, mapping them to Desmos, and reading or working out the value the question asks for.

Difficulty: Advanced
Estimated time: 50 minutes
Domains: Algebra

What you’ll learn

Spot a repeated chunk, a whole expression that the system treats as one quantity.
Give each chunk a short temporary name.
Put two named chunks on the calculator's $x$ - and $y$ -axes.
Read the chunk, or mix of chunks, that the question asks for from where the new lines meet.
Work back to the original $x$ and $y$ when that's the clearest way to the answer.
Choose between graphing the smaller system, eliminating by hand, and solving the original system.

Prerequisites

Solve systems at intersections0%
Graph two equations, read their intersection, and add or subtract equations when that eliminates a variable.

Why this matters on the SAT

Solve the smaller system hiding inside the larger one

Some SAT systems look hard only because the same expressions keep showing up. You could expand every set of parentheses, and that works, but it makes long equations and gives you more places to drop a sign. The faster move is to rename the repeats. A much smaller system appears.

SAT example

\begin{aligned} (3x+2y)+2(2x-3y)&=7\\[1.4em] 5(3x+2y)-(2x-3y)&=29 \end{aligned}

What is the value of $3x+2y$ ?

Fast Desmos solution

Two expressions repeat, so give them short names:

U=3x+2y

and

V=2x-3y.

Now the system is much smaller. We'll call it the reduced system:

\begin{aligned} U+2V&=7\\[1.4em] 5U-V&=29. \end{aligned}

Desmos graphs in $x$ and $y$ , so let calculator $x$ stand for $U$ and calculator $y$ stand for $V$ . Type:

x+2y=7

5x-y=29

The lines meet near

(5.91,0.545).

Calculator $x$ is $U$ , and $U$ is $3x+2y$ , the value the question asks for. The decimal points to D, and a little algebra confirms it exactly. Multiply $5U-V=29$ by $2$ , then add $U+2V=7$ . The $V$ -terms cancel:

11U=65,

U=3x+2y=\boxed{\frac{65}{11}}.

The answer is D.

Choice A, $\frac{6}{11}$ , is the value of $V$ , the other coordinate. Read the wrong one and you'd land right on it.

You never had to find the original $x$ and $y$ . The question asked for the first chunk, so the first coordinate was enough.

Calculator loads as you approach

Calculator x stands for 3x + 2y from the problem, and calculator y stands for 2x - 3y.

When should you use this method?

Renaming pays off when a system is built out of the same few pieces.

Rename the repeats when…

The same two expressions keep showing up, like $(3x+2y)$ and $(2x-3y)$ in the example above.
Most of the work around those expressions only multiplies, adds or subtracts them, and expanding would make the equations longer before they get shorter.
The question asks for one of the expressions, a mix of them like $U+V$ , or something you can work out from them.

Take a shorter path when…

The original equations are already short, like $2x+y=7$ and $x-y=2$ . Graph them the usual way, as in Solve systems at intersections.
After renaming, adding or subtracting the reduced equations gives the answer right away, like $U+V=10$ and $U-V=4$ , where adding gives $2U=14$ . Do that by hand instead of graphing.

One quick test before you rename: expand in your head. If a variable cancels on the spot, the way $(x+y)+(x-y)=10$ turns into $2x=10$ , skip the renaming.

Check your understanding:

A system keeps repeating $4x-y$ and $x+3y$ . What would you name them?

Name whole chunks before you expand

Look at this system:

\begin{aligned} 3(5x+2y)-2(x-y)&=14\\[1.4em] (5x+2y)+4(x-y)&=23. \end{aligned}

The same two expressions sit inside the parentheses in both equations, so name them:

U=5x+2y

and

V=x-y.

Copy each chunk exactly as written, signs and all. Now the system reads

\begin{aligned} 3U-2V&=14\\[1.4em] U+4V&=23. \end{aligned}

The original $x$ and $y$ haven't gone anywhere. They're tucked inside $U$ and $V$ , and you can get back to them if you need to.

Check your understanding:

In the first equation above, is the first chunk $5x$ , $5x+2y$ , or $3(5x+2y)$ ?

Common mistake:

It’s tempting to expand everything first and look for the pattern afterward. But once the parentheses are gone, the repeats are much harder to see. Name the chunks while the parentheses still show them to you.

Put the chunks on the calculator's axes

Desmos graphs in $x$ and $y$ , not $U$ and $V$ . So before you type, decide which chunk goes on which axis. This is your axis map:

\begin{aligned} U&\longrightarrow \text{calculator }x,\\[1.4em] V&\longrightarrow \text{calculator }y. \end{aligned}

Here's the part that trips people up. Once you start typing, the calculator's $x$ and $y$ are your chunks, not the $x$ and $y$ from the problem. Keep using $U$ and $V$ on scratch paper, and write the map down, so the two sets of letters stay separate.

Say the reduced system is

\begin{aligned} 2U+V&=11\\[1.4em] U-V&=1. \end{aligned}

Type:

2x+y=11

x-y=1

The lines meet at $(4,3)$ . Using the map, that means

U=4

and

V=3.

Try it yourself:

Check on the graph that the lines meet at $(4,3)$ . If they aren’t showing, type both reduced equations. Then use those values to work out $2U-V$ .

Check your understanding:

The lines meet at $(U,V)=(4,3)$ . What’s $2U-V$ , and how can you check the chunk values?

Calculator loads as you approach

The lines meet at (4, 3), so U = 4 and V = 3.

Check what the question asks before you do more

Once you have $U$ and $V$ , go back to the question and read exactly what it wants. The same point,

(U,V)=(4,3),

answers different questions in different ways:

\begin{aligned} U&=4,\\[1.4em] V&=3,\\[1.4em] U+V&=7,\\[1.4em] 2U-V&=5. \end{aligned}

As soon as you have what the question asks for, stop. If it's one chunk or a simple mix of chunks, you don't need the original $x$ and $y$ at all.

Check your understanding:

A reduced system puts $U=7x-5y$ on the calculator $x$ -axis and $V=4x+3y$ on the calculator $y$ -axis. The lines meet at $(2,-3)$ . What is the value of $4x+3y$ ?

Common mistake:

When the question asks for one expression, don’t answer with the whole point. The point holds both chunk values, and the answer is only the one the question names.

Decide whether to recover the original variables

Sometimes the question asks about the original $x$ and $y$ . Before you solve for them, ask yourself: can I build the target out of $U$ and $V$ ?

Say the chunks are

U=2x-y

and

V=x+3y,

and the question asks for $3x+2y$ . Try adding the chunks:

U+V=3x+2y.

That's the target. So if $U=4$ and $V=5$ , then $3x+2y=4+5=9$ , and you never need $x$ or $y$ on their own.

When you can't see a quick way to build the target, go the other way. Set each chunk equal to its value and solve that recovery system for $x$ and $y$ . Recovering the original variables is a choice, not a required step, so save it for when it's the clearest way left.

Check your understanding:

You’ve found the values of $U=2x-y$ and $V=x+4y$ . Which of these can you answer right away: $U$ , $U+V$ , or $5x-2y$ ?

Example: Reduce, recover, and answer

Worked example

The solution to the system of equations

\begin{aligned} 4(2x-y)-(x+4y)&=11\\[1.4em] 4(2x-y)+(x+4y)&=65 \end{aligned}

is $(x,y)$ . Which choice is equal to $5x-2y$ ?

A
$\frac{236}{9}$
B
$\frac{182}{9}$
C
$236$
D
$\frac{73}{2}$

Step 1

Name the repeated chunks

Two expressions repeat, so let

U=2x-y

and

V=x+4y.

The system becomes

\begin{aligned} 4U-V&=11\\[1.4em] 4U+V&=65. \end{aligned}

Step 2

Solve the reduced system

Look at the $V$ -terms: one is $-V$ and the other is $+V$ . Adding the equations cancels them, which is faster than graphing:

8U=76,

U=\frac{19}{2}.

To find $V$ , subtract the first equation from the second:

2V=54,

V=27.

Step 3

Recover the original variables

It's tempting to stop here. Adding the chunk values gives $\frac{19}{2}+27=\frac{73}{2}$ , which is choice D. But that's $U+V$ , and the question asks for $5x-2y$ .

Could you build $5x-2y$ out of $U$ and $V$ ? You can, but it takes $\frac{22}{9}U+\frac{1}{9}V$ , which is hard to spot. So go back to what the chunks mean:

\begin{aligned} 2x-y&=\frac{19}{2}\\[1.4em] x+4y&=27. \end{aligned}

This recovery system is small, so solve it by hand. The first equation gives

y=2x-\frac{19}{2}.

Substitute that into the second equation:

\begin{aligned} x+4\left(2x-\frac{19}{2}\right)&=27\\[1.4em] x+8x-38&=27\\[1.4em] 9x&=65\\[1.4em] x&=\frac{65}{9}. \end{aligned}

Then

y=2\left(\frac{65}{9}\right)-\frac{19}{2}=\frac{260}{18}-\frac{171}{18}=\frac{89}{18}.

Step 4

Answer the question that was asked

Now put $x$ and $y$ into the target:

\begin{aligned} 5x-2y &=5\left(\frac{65}{9}\right)-2\left(\frac{89}{18}\right)\\[1.4em] &=\frac{325}{9}-\frac{89}{9}\\[1.4em] &=\frac{236}{9}. \end{aligned}

The answer is A.

Notice that you solved two systems here, and they mean different things. The first gave you $U$ and $V$ . The second gave you the original $x$ and $y$ .

Common mistake:

Don’t treat $\left(\frac{19}{2},27\right)$ as the original $(x,y)$ . Those are the values of the chunks $U$ and $V$ , not of $x$ and $y$ .

Pick the shortest way to solve the smaller system

Renaming gives you a smaller system, but you don't have to graph it. Look at it first. Take

\begin{aligned} 3U+2V&=16\\[1.4em] 3U-2V&=8, \end{aligned}

where the question asks for $U$ . The $V$ -terms are opposites, so add the equations:

6U=24,

so $U=4$ . Opening Desmos here would add setup and tell you nothing new.

Graph the reduced system when nothing cancels that fast, like $U+2V=7$ and $5U-V=29$ from the first example, where you'd have to multiply an equation before anything cancels.

Check your understanding:

A system becomes $5U+2V=31$ and $3U-2V=9$ , and the question asks for $U$ . What’s the shortest next step?

Finish the solution

The first reduced equation is already in the calculator. Type the second one, then answer with the coordinate that belongs to the chunk the question asks for.

Read the target chunk

Finish the solution

The solution to the system

\begin{aligned} 3(4x-y)+2(x+3y)&=25\\[1.4em] 5(4x-y)-(x+3y)&=7 \end{aligned}

is $(x,y)$ . What is the value of $4x-y$ ?

First steps

Let $U=4x-y$ and $V=x+3y$ .
The reduced system is $3U+2V=25$ and $5U-V=7$ .
Put $U$ on calculator $x$ and $V$ on calculator $y$ .

Finish it

Your answer

Calculator loads as you approach

The first reduced equation is in. Add the second, then read the coordinate that belongs to U.

Practice problems

Now it's your turn. These three problems each call for a different move, so look at the equations before you pick a method.

Simplify before naming chunks

Practice problem

The solution to the system

\begin{aligned} 2(3x-y)+(x+2y)&=13\\[1.4em] (3x-y)-3(x+2y)&=-4 \end{aligned}

is $(x,y)$ . What is the value of $x$ ?

Your answer

Calculator loads as you approach

Look at the equations before you pick a method.

Map a reduced system independently

Practice problem

The solution to the system of equations

\begin{aligned} 3(2x+y)+2(x-4y)&=31\\[1.4em] 2(2x+y)-3(x-4y)&=-1 \end{aligned}

is $(x,y)$ . What is the value of $2x+y$ ?

Your answer

Calculator loads as you approach

Rename, put the chunks on the axes, then read the right coordinate.

Eliminate, then recombine the chunks

Practice problem

The solution to the system of equations

\begin{aligned} 7(2x-3y)-5(x+4y)&=9\\[1.4em] 7(2x-3y)+5(x+4y)&=229 \end{aligned}

is $(x,y)$ . What is the value of $11(x+y)$ ?

Your answer

Calculator loads as you approach

Rename, solve for the chunks, then build the target from them.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Look for whole expressions that repeat without changing.
Give each one a short name, like $U$ and $V$ .
Put the first chunk on calculator $x$ and the second on calculator $y$ .
Graph the reduced system, or eliminate by hand when a chunk cancels right away.
Stop as soon as you have what the question asks for, whether that's one chunk or a simple mix of chunks.
Recover $x$ and $y$ only when you can't quickly build the target from the chunks and recovery is the clearer route.
Check the chunk values in both reduced equations, then work out the exact expression the question names.

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