Isolate a hidden expression

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Difficulty: Advanced
Estimated time: 50 minutes
Domains: Algebra
Techniques: Hidden-expressionSubstitutionSystemsCoordinate-mappingIsolationMethod-choice

What you’ll learn

Recognize a complete expression that repeats inside a system.
Give repeated chunks short temporary names.
Map two renamed chunks to the calculator axes and solve the reduced system.
Read the requested chunk or combination from the reduced values.
Recover the original variables only when that is the clearest route.
Choose between reduced graphing, direct elimination, and simplifying 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 difficult only because the same expressions appear repeatedly. A chunk is a complete repeated expression that the system treats as one quantity. Expanding every parenthesis works, but it can create long equations and more chances for a sign error. Renaming the chunks reveals a much smaller system.

SAT example

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

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

A
$\frac{6}{11}$
B
$\frac{53}{11}$
C
$\frac{77}{11}$
D
$\frac{65}{11}$

Fast Desmos solution

Let $U=3x+2y$ and $V=2x-3y$ . The system becomes

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

Map $U$ to calculator $x$ and $V$ to calculator $y$ . The reduced lines intersect near $(5.91,0.545)$ , so the first coordinate points to choice D.

Confirm the target exactly. Multiply $5U-V=29$ by $2$ , then add $U+2V=7$ :

11U=65.

Therefore,

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

The answer is D. The requested chunk is known, so the original $x$ and $y$ are unnecessary.

Calculator loads as you approach

Calculator x represents 3x + 2y from the problem, while calculator y represents 2x - 3y.

When should you use this method?

Look for the same complete expressions appearing more than once. Then inspect what surrounds those chunks and what the question asks for.

Reduce the system when…

The chunks repeat: most outside work only multiplies, adds, or subtracts the same two expressions.
The target aligns: the question asks for one chunk, a simple combination of chunks, or a value easily recovered from them.

Use direct algebra when…

Terms cancel immediately: adding or subtracting the equations isolates the target in one step.
The original system collapses: expansion produces one-step equations, so new variables would add work.

Inspect the structure before choosing a tool. A valid method is not always the shortest method.

Check your understanding:

A system repeatedly contains $4x-y$ and $x+3y$ . What two temporary names would keep the structure visible?

1. Name complete chunks before expanding

\begin{aligned} 3(5x+2y)-2(x-y)&=14\\ (5x+2y)+4(x-y)&=23, \end{aligned}

the useful chunks are

U=5x+2y

and

V=x-y.

Keep every sign and coefficient inside the chunk exactly as written. The reduced system is

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

The original variables disappear from the outer system, but their meaning remains in the definitions of $U$ and $V$ .

Check your understanding:

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

Common mistake:

Expanding first and trying to rediscover the pattern later. Name repeated chunks while the parentheses still make them easy to see.

2. Map the chunks to calculator axes

On scratch paper, letters such as $U$ and $V$ keep the new quantities distinct from the original variables. For a two-dimensional Desmos graph, make an explicit axis map:

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

Suppose the reduced system is

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

Enter 2x+y=11 and x-y=1. Desmos shows $(4,3)$ . Under the axis map, that means $U=4$ and $V=3$ .

Try it yourself:

If the graph is blank, enter both reduced equations. Confirm the intersection is $(4,3)$ , then evaluate $2U-V$ from those chunk values.

Check your understanding:

The reduced intersection is $(U,V)=(4,3)$ . What is $2U-V$ , and how can you verify the chunk values?

Calculator loads as you approach

The intersection (4, 3) means U = 4 and V = 3 under the stated axis map.

3. Read the target before doing more work

After finding $U$ and $V$ , return to the exact expression in the question. For $(U,V)=(4,3)$ , different targets produce

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

Stop as soon as you have the requested quantity. Do not recover the original variables if the target is already one chunk or a simple combination of the chunks.

Check your understanding:

A reduced system maps $U=7x-5y$ to calculator $x$ and $V=4x+3y$ to calculator $y$ . The intersection is $(2,-3)$ . What is $4x+3y$ ?

Common mistake:

Reporting the reduced ordered pair when the SAT asks for only one expression. The intersection stores the chunk values; it is not automatically the final response.

4. Decide whether to recover the original variables

When the target uses the original $x$ and $y$ , first ask whether it is easy to rewrite the target using $U$ and $V$ . If the rewrite is obvious, compute directly from the chunk values. If it would require another complicated derivation, return to the chunk definitions and solve a recovery system.

For example, if $U=2x-y$ and $V=x+3y$ , then

U+V=3x+2y.

If $U=4$ and $V=5$ , the target $3x+2y$ is $4+5=9$ . There is no need to recover $x$ and $y$ separately. Recovery is a choice, not an automatic step.

Check your understanding:

If $U=2x-y$ and $V=x+4y$ are already known, which targets are immediate: $U$ , $U+V$ , or $5x-2y$ ?

5. Example: Reduce, recover, and answer

Worked example

The solution to the system of equations

$4$

$(2x-y)$

$-$

$(x+4y)$

$=11$

$4$

$(2x-y)$

$+$

$(x+4y)$

$=65$

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

Let $U=2x-y$ and $V=x+4y$ . The system becomes

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

Step 2

Solve the reduced system

The reduced equations already have opposite $V$ -terms, so elimination is shorter than graphing. Add them:

8U=76,

which gives $U=\frac{19}{2}$ . Subtract the first reduced equation from the second:

2V=54,

so $V=27$ .

Step 3

Recover the original variables

The target is not an immediate combination of $U$ and $V$ , so return to the chunk definitions:

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

Solve the recovery system algebraically:

\begin{aligned} y&=2x-\frac{19}{2},\\ x+4\left(2x-\frac{19}{2}\right)&=27,\\ 9x&=65,\\ x&=\frac{65}{9},\\ y&=\frac{89}{18}. \end{aligned}

Step 4

Evaluate only the requested expression

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

The answer is A. The reduced values belong to $U$ and $V$ ; the recovery system gives the original $x$ and $y$ .

Common mistake:

Using $\left(\frac{19}{2},27\right)$ as the original $(x,y)$ . Those are the renamed chunk values, not the original variable values.

6. Choose the shortest route after renaming

Renaming does not force you to graph every reduced system. Read the reduced intersection when the equations do not cancel conveniently. Use hand elimination when adding or subtracting isolates the target. Recover the original variables only when the target is not quickly expressible from the chunks. Solve the original system when it is already short.

For example, if

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

then adding gives $6U=24$ , so $U=4$ . Opening Desmos would add setup without adding clarity.

Check your understanding:

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

Finish the solution

The first reduced equation is already graphed. Apply the axis map, enter the second reduced equation, and submit the coordinate mapped to the requested chunk.

Read the target chunk

Finish the solution

The solution to the system

\begin{aligned} 3(4x-y)+2(x+3y)&=25\\ 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 $5U-V=7$ .
Map $U$ to calculator $x$ and $V$ to calculator $y$ .

Finish it

Your answer

Calculator loads as you approach

The first reduced equation is entered. Add the second and read the coordinate mapped to U.

Practice problems

SAT practice problems

Inspect the original structure first. Then choose direct simplification, reduced graphing, elimination, recombination, or recovery.

Simplify before naming chunks

Practice problem

The solution to the system

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

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

Your answer

Calculator loads as you approach

Inspect the original equations before deciding whether renamed chunks or graphing would shorten the work.

Map a reduced system independently

Practice problem

The solution to the system

\begin{aligned} 3(2x+y)+2(x-4y)&=31\\ 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

Name the repeated chunks, map them to calculator x and y, and read the coordinate that represents the target.

Eliminate, then recombine the chunks

Practice problem

The solution to the system

\begin{aligned} 7(2x-3y)-5(x+4y)&=9\\ 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 the chunks, use graphing or elimination for the reduced system, and rewrite the target from those chunk values.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Spot: Find complete expressions that repeat without changing.
Name: Give the chunks temporary names such as $U$ and $V$ .
Map: Use calculator $x$ for the first chunk and calculator $y$ for the second.
Solve: Graph the reduced system or use elimination if it is clearly shorter.
Read: Stop if the target is already one chunk or a simple chunk combination.
Recover: Return to the definitions only when recovery is the clearer route.
Verify: Check reduced values, then evaluate the exact expression requested.

Return to the Desmos for the SAT course to review earlier techniques or choose another lesson.

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