REVIEWING THE MODELS FOR SOLVING EQUATIONS

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REVIEWING THE MODELS FORSOLVING EQUATIONS

• Robert Yen
• Hurlstone Agricultural High School

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OVERVIEW

• To review and compare 3 models for teaching
  equations
• Students often have trouble solving equations
  because their teachers teach only one method, the
  method they were taught themselves

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OVERVIEW

• To discuss and share ideas and classroom
  experiences on teaching equations

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Model 1GUESS, CHECK AND IMPROVE

• Alternative names
• Guess and check
• Guess, check and refine
• Trial and error
• By inspection

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Model 1GUESS, CHECK AND IMPROVE

• Description
• The name explains the method guess the solution,
  test it, make a better guess, keep testing
• It is a process that cycles, is repetitive

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Model 1GUESS, CHECK AND IMPROVE

• Example 1
• x 5 40
• By inspection, x 35
• because 35 5 40.

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Model 1GUESS, CHECK AND IMPROVE

• Example 2
• 10
• By inspection, d 30
• because 10.

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Model 1GUESS, CHECK AND IMPROVE

• Example 3
• 4x 5 57

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Model 1GUESS, CHECK AND IMPROVE

• Example 3
• 4x 5 57

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Model 1GUESS, CHECK AND IMPROVE

• Example 3
• 4x 5 57

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Model 1GUESS, CHECK AND IMPROVE

• Example 3
• 4x 5 57

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Model 1GUESS, CHECK AND IMPROVE

• Example 3
• 4x 5 57

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Model 1GUESS, CHECK AND IMPROVE

• Example 3
• 4x 5 57
• x 13.

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6

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Model 1GUESS, CHECK AND IMPROVE

• Example 4
• 3x 4 x 6
• x -5

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Model 1GUESS, CHECK AND IMPROVE

• What the syllabus says (p.86, PAS4.4)
• Five models have been proposed to assist
  students with the solving of simple equations ...
  Model 4 uses a substitution approach. By trial
  and error a value is found that produces equality
  for the values on either side of the equation
  (this highlights the variable concept).

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Model 1GUESS, CHECK AND IMPROVE

• Advantages
• ???

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Model 1GUESS, CHECK AND IMPROVE

• Advantages
• Reinforces aim of solving equations and algebraic
  concepts of unknown, variable
• Reinforces checking of solutions
• Simple to understand and apply
• Feedback on partial solutions, homing in on
  answer, unlike algebraic methods where one
  careless error will undermine the solution process

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Model 1GUESS, CHECK AND IMPROVE

• Advantages
• Being repetitive, can be performed via
  technology spreadsheet, graphics calculator
• With better guesses, solution can be found
  quickly
• Can improve students computation skills and
  number sense

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Model 1GUESS, CHECK AND IMPROVE

• Disadvantages
• ???

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Model 1GUESS, CHECK AND IMPROVE

• Disadvantages
• Guesswork is not an elegant method
• Harder to apply for more complex equations (such
  as x on both sides)
• May be hard to test values that are large or
  negative
• More time-consuming if guesses are bad

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Model 2 BALANCING

• Alternative name
• Doing the same thing to both sides (of the
  equation)

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Model 2 BALANCING

• Description
• The traditional algebraic method
• Models the equation as balance scales, upon which
  the same inverse operations are performed on both
  sides to create equivalent equations until it
  simplifies to x ___

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Model 2 BALANCING

• Description
• Invented by Arab mathematician
• al-Khwarizmi in AD 825, who wrote
• Hisab al-jabr wal-muqabalah
• The science of restoration and cancellation
• al-jabr restoration by balancing,
• from which we get the name algebra
• muqabalah cancellation of terms

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Model 2 BALANCING

• Description
• From al-Khwarizmis name, we get the name
  algorithm
• Can be demonstrated using concrete objects such
  as cups, envelopes, counters, coins or coloured
  dots
• Or coloured plastic bottle caps (see session by
  Kevin Fuller tomorrow _at_ 2 pm)

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Model 2 BALANCING

• Example 1 (A concrete model)
• 2x 7 9

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Model 2 BALANCING

• Example 1 (A concrete model)
• 2x 7 9
• Subtract 7 coins from both sides

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Model 2 BALANCING

• Example 1
• 2x 7 9

Place the remaining coins into two equal rows
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Model 2 BALANCING

• Example 1
• 2x 7 9
• Divide both sides by 2

Place the remaining coins into two equal rows
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Model 2 BALANCING

• Example 1
• 2x 7 9
• x 1
• Check 2(1) 7 9

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Model 2 BALANCING

• Example 1 algebraically
• 2x 7 9
• 2x 7 7 9 7
• 2x 2
• 2x/2 2/2
• x 1

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Model 2 BALANCING

• Example 2 (Another concrete model)
• 3x 2 x 10

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Model 2 BALANCING

• Example 2 (Another concrete model)
• 3x 2 x 10
• Subtract x from both sides

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Model 2 BALANCING

• Example 2
• 3x 2 x 10

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Model 2 BALANCING

• Example 2
• 3x 2 x 10
• Subtract 2 from both sides

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Model 2 BALANCING

• Example 2
• 3x 2 x 10

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Model 2 BALANCING

• Example 2
• 3x 2 x 10
• Divide both sides by 2

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Model 2 BALANCING

• Example 2
• 3x 2 x 10
• x 4
• Check 3(4) 2 14
• 4 10 14

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Model 2 BALANCING

• Example 2 algebraically
• 3x 2 x 10
• 3x 2 x x 10 x
• 2x 2 10
• 2x 2 2 10 2
• 2x 8
• 2x/2 8/2
• x 4

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Model 2 BALANCING

• Note that the appropriate
• inverse operations must be identified
• and performed in the correct order.
• Aim to have x on its own
• on the LHS of the equation
• x ___

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 1 uses a two-pan balance and objects
  such as coins or centicubes. A light paper
  wrapping can hide a mystery number of objects
  without distorting the balances message of
  equality.

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 2 uses small objects (all the same)
  with some hidden in containers to produce the
  unknowns or mystery numbers, eg place the
  same number of small objects in a number of paper
  cups and cover them with another cup. Form an
  equation using the cups and then remove objects
  in equal amounts from each side of a marked
  equals sign.

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 3 uses one-to-one matching of terms on
  each side of the equation, eg
• 3x 1 2x 3
• x x x 1 x x 2 1
• By one-to-one matching and cancelling

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 3 uses one-to-one matching of terms on
  each side of the equation, eg
• 3x 1 2x 3
• x x x 1 x x 2 1
• By one-to-one matching and cancelling
• x x x 1 x x 2 1

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 3 uses one-to-one matching of terms on
  each side of the equation, eg
• 3x 1 2x 3
• x x x 1 x x 2 1
• By one-to-one matching and cancelling
• x x x 1 x x 2 1

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 3 uses one-to-one matching of terms on
  each side of the equation, eg
• 3x 1 2x 3
• x x x 1 x x 2 1
• By one-to-one matching and cancelling
• x x x 1 x x 2 1

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 3 uses one-to-one matching of terms on
  each side of the equation, eg
• 3x 1 2x 3
• x x x 1 x x 2 1
• By one-to-one matching and cancelling
• x x x 1 x x 2 1

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Model 2 BALANCING

• What the syllabus says (p.86, PAS4.4)
• Model 3 uses one-to-one matching of terms on
  each side of the equation, eg
• 3x 1 2x 3
• x x x 1 x x 2 1
• By one-to-one matching and cancelling
• x 2.

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Model 2 BALANCING

• Advantages
• ???

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Model 2 BALANCING

• Advantages
• Powerful and elegant logical method
• Works for all types of equations, including those
  with x on both sides
• If done correctly, solution emerges quickly
• Reinforces algebraic concepts of balance and
  equivalence of expressions (by cancelling and
  simplifying)

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Model 2 BALANCING

• Disdvantages
• ???

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Model 2 BALANCING

• Disdvantages
• Harder to model 6 2x 14,
• x 2 10 with concrete objects how do you
  represent subtraction, division or squaring of
  objects?
• Difficult for some students to understand,
  conceptualise, reason

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Model 2 BALANCING

• Disdvantages
• Some students have trouble knowing which inverse
  operation to perform first
• Lines of working can appear complicated and messy
• Students often dont know what they are actually
  doing or why they are doing it

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Model 2 BALANCING

• Example 3 (a model for negatives)
• 6 2x 14
• Subtract 6 from both sides
• 6 2x 6 14 6

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Model 2 BALANCING

• Example 3
• 6 2x 14
• -2x 8
• Divide both sides by 2

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Model 2 BALANCING

• Example 3
• 6 2x 14
• -x 4
• Take the negative of both sides

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Model 2 BALANCING

• Example 3
• 6 2x 14
• x -4
• Check 6 2(-4) 14

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Model 3 BACKTRACKING

• Alternative names
• Un-doing or unpacking
• Reverse flowchart

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Model 3 BACKTRACKING

• Description
• An alternative algebraic model
• To undo the operations performed on the variable
  on one side of an equation, inverse operations
  are performed on the other side of the equation

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Model 3 BACKTRACKING

• Description
• Can be demonstrated using a flowchart and reverse
  flowchart
• Models the equation as a sequence of operations
  on a variable that are undone by a sequence of
  inverse operations that backtrack to the
  variable

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Model 3 BACKTRACKING

• Example 1
• 2x 7 9
• Use a flowchart to go from x to 2x 7

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Model 3 BACKTRACKING

• Example 1
• 2x 7 9
• Use a flowchart to go from x to 2x 7

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Model 3 BACKTRACKING

• Example 1
• 2x 7 9
• But 2x 7 9

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Model 3 BACKTRACKING

• Example 1
• 2x 7 9
• To get back to x, undo the operations
• using a reverse flowchart

To undo 7, we 7
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Model 3 BACKTRACKING

• Example 1
• 2x 7 9
• To get back to x, undo the operations
• using a reverse flowchart

To undo ? 2, we ? 2
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Model 3 BACKTRACKING

• Example 1
• 2x 7 9
• To get back to x, undo the operations
• using a reverse flowchart
• x 1

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Model 3 BACKTRACKING
With backtracking, we (inverse) operate on one
side only

• Example 1 algebraically
• 2x 7 9
• 2x 9 7
• 2
• x 2/2
• x 1

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Model 3 BACKTRACKING

• Inverse operations are
• performed in reverse order.
• For example,
• to undo putting on our socks,
• then our shoes,
• we take off our shoes first,
• then take off our socks.

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Model 3 BACKTRACKING

• Example 2
• Use a flowchart to go from y to

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Model 3 BACKTRACKING

• Example 2
• Use a flowchart to go from y to

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Model 3 BACKTRACKING

• Example 2
• Use a flowchart to go from y to

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Model 3 BACKTRACKING

• Example 2
• Use a reverse flowchart to backtrack to y

To undo ? 5, we ? 5
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Model 3 BACKTRACKING

• Example 2
• Use a reverse flowchart to backtrack to y

To undo 3, we 3
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Model 3 BACKTRACKING

• Example 2
• Use a reverse flowchart to backtrack to y
• y 7

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Model 3 BACKTRACKING

• Example 2 algebraically
• y 3 2 ? 5
• 10
• y 10 3
• y 7

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Model 3 BACKTRACKING

• What the syllabus says (p.86, PAS4.4)
• Model 5 uses backtracking or a reverse flow
  chart to unpack the operations and find the
  solution.

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Model 3 BACKTRACKING

• Advantages
• ???

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Model 3 BACKTRACKING

• Advantages
• Only un-doing one side of equation less working
• For some students, this method is more intuitive
  its what we do when we solve an equation
  mentally
• Conceptually easier to understand than balancing

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Model 3 BACKTRACKING

• Advantages
• Algebraic working consistent with balancing
  method
• If done correctly, solution emerges quickly
• Reinforces skills in algebraic notation and
  generalising formulas

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Model 3 BACKTRACKING

• Disdvantages
• ???

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Model 3 BACKTRACKING

• Disdvantages
• Takes longer to teach as it requires careful
  practice with flowcharts
• Does not work if x on both sides of equation, eg
  3x 2 x 10
• Harder to model if x is not in the first term, eg
  6 2x 14

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So which is the best method to use?
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So which is the best method to use?

• All three methods have merit and can be used
  together in the classroom
• Depends on the class
• Guess, check and improve is good for starting the
  topic, and leads to the idea of inverse
  operations and algebraic methods

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• Backtracking is a useful tool for students who
  struggle with algebra, again convenient for
  introducing inverse operations
• Once students are confident with the algebra,
  introduce harder equations that require balancing
  to simplify the equation first, for example,
• x on both sides
• x is not in the first term
• equations with brackets

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x on both sides

• 3x 2 x 10

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x on both sides

• 3x 2 x 10
• 3x 2 x 10

Use an inverse operation to remove the x from the
RHS
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x on both sides

• 3x 2 x 10
• 3x 2 x 10
• 2x 2 10

Simplifies to a 2-step equation proceed by
backtracking or balancing
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x on both sides

• 3x 2 x 10
• 3x 2 x 10
• 2x 2 10
• 2x 10 2

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x on both sides

• 3x 2 x 10
• 3x 2 x 10
• 2x 2 10
• 2x 10 2
• 8

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x on both sides

• 3x 2 x 10
• 3x 2 x 10
• 2x 2 10
• 2x 10 2
• 8

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x on both sides

• 3x 2 x 10
• 3x 2 x 10
• 2x 2 10
• 2x 10 2
• 8
• x 4

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x is not in the first term

• 6 2x 14

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x is not in the first term

• 6 2x 14
• -2x 6 14

If backtracking, rewrite so that x is in the
first term
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x is not in the first term

• 6 2x 14
• -2x 6 14
• -2x 14 6

Proceed by backtracking or balancing
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x is not in the first term

• 6 2x 14
• -2x 6 14
• -2x 14 6
• 8

Proceed by backtracking or balancing
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x is not in the first term

• 6 2x 14
• -2x 6 14
• -2x 14 6
• 8

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x is not in the first term

• 6 2x 14
• -2x 6 14
• -2x 14 6
• 8
• x -4