Algebra I: Section 3



CC MATH I STANDARDS UNIT 3

4.6 Solving Equations with VARIABLE ON BOTH SIDES: PART 1

WARM UP: Solve for the given variable.

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INTRODUCTION: GUESS VALUES and CHECK by Substitution

Example: 5x = x – 12

|GUESS: |CHECK: |

|#1: x = 4 |5(4) = 20 and 4 – 12 = -8 |NO; 20 and -8 are 28 apart |

|#2: x = 6 |5(6) = 30 and 6 – 12 = -6 |NO; 30 and 06 are 36 apart |

|#3: x = 2 |5(2) = 10 and 2 – 12 = -10 |NO; 10 and -10 are 20 apart |

|#4: x = 0 |5(0) = 0 and 0 – 12 = -12 NO |NO; 0 and -12 are 12 apart |

|#5: x = -2 |5(-2) = -10 and -2 – 12 = -14 NO |NO; -10 and -14 are 4 apart |

|#6: x = -3 |5(-3) = -15 and -3 – 13 = -15 |YES; x = -3 is a solution |

1) 2a + 7 = 3a + 4

2) 4 + 6y – 6 = 6y – 2

3) 2z + 2 = -6 + 2z

TYPES OF ANSWERS IN VARIABLE ON BOTH SIDES PROBLEMS

• Variables on both sides can create 3 different type of solution scenarios

|Type #1: |Type #2: ALL NUMBERS |Type #3: |

|ONE SOLUTION |(Identity) |NO SOLUTION |

|EXACTLY ONE number can be used as an answer |ANY NUMBER (pos, neg, fraction, whole, etc) can be |NO POSSIBLE NUMBER can be used as an answer |

| |used as an answer | |

|Example: | |Example: |

|2a + 7 = 3a + 4 |Example: |2z + 2 = -6 + 2z |

|Solution: |4 + 6y – 6 = 6y – 2 |Solution: |

|a = 3 is only answer |Solution: |No value will ever work |

| |y = 0, -1, 3 ALL WORK | |

What is the process for solving with VARIABLES on BOTH SIDES?

You don’t want guess and check if the answer is NO SOLUTION, DECIMAL, or FRACTION answer. The following steps will lead you to the correct solution.

Step #1: If needed, SIMPLIFY each side of equation separately

• Distribute Property AND Combine any like terms

Step #2: MOVE by adding or subtracting VARIABLE LIKE TERMS to one side of the equation

Step #3: SOLVE the remaining equation (SADMEP)

How do we identify when solving what type of equations have

ONE Solution, NO Solution, or All NUMBERS?

• Always check which type of solution your equation satisfies.

|Example #1: |Example #2: |Example #3: |

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BEGINNING EXAMPLES of MOVING VARIABLE TERMS:

1) a + 9 = 4a

2) -5b = 7b + 36

3) 9x = -12 + 9x

4) 4x – 15 = 7x

5) 7r + 56 = 15r

6) 40 + 10y = - 10y

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INTERMEDIATE EXAMPLES of MOVING VARIABLE TERMS:

It doesn’t matter where you move the variable if you watch the positive and negative signs

GENERAL RULE OF THUMB: Move the SMALLER Variable to the BIGGER VARIABLE

1) 10x + 3 = 8x + 9

2) 7 – 2z = 22 + 3z

3) 3y - 13 = 3y + 16

4) 7r - 6 = - 3r + 44

5) 5x – 9 = -9 + 5x

6) 13 – 3y = 6y - 59

7) – 5y + 15 = -4y – 12

8) -2z + 12 = - 12 + 2z

9) - 6x + 8 = - 5 - 6x

ADVANCED EXAMPLES of MOVING VARIABLE TERMS:

SIMPLIFY EACH SIDE BEFORE MOVING TERMS

Use a separate piece of paper as needed.

1) 7p – 2 + 3p = 8p – 1

2) 7r + 10 = -10 + 8r – 22

3) 2m + 5 = 5m – 35 – 3m

4) 3r + 3 – 5 = 9r – 2 – 6r

5) -3s + 8 + 8s = 7s – 2

6) 4q + 6 = 5q – 42 + 7q

7) 13t – 9t + 80 = 4t + 100 – 20

8) 40c + 8 – 13 = 5 + 40c

9) 13x – 21x + 17 = -3x – 5 – 8

10) 12 – 7x – 2x = 8x + 12 – 17x

11) 3 + 3t – 7 = 3t + 4

12) 8 – 12z = 15 – 3z + 20

13) 15m – 7m + 11 = 12 – 5m + 2 + 7 m

14) 8 – 3n – 4n + 16 = 5n – 6 + 2n – 12

15) 8p + 9 = 3 + 8p + 2

16) 7r – 4r + 12 = -6r + 12 + 9r

CC MATH I STANDARDS UNIT 3

4.6 Solving Equations with VARIABLE ON BOTH SIDES: PART 2

WARM UP: Solve for the given variable.

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REVIEW: SIMPLIFYING

#1) 7(3x + 6) – 5x

#2) 12z – 7 + 9z – 8

#3) 6y – 4(5 – 7y)

#4) – 5r + 9 – 3r – 7r

#5) 5 – 2 (8 + 11n)

#6) 3(x + 7) + 4(2x – 5)

VARIABLE ON BOTH SIDES: SOLUTION TYPES

What are the 3 types of solutions you can have in an equation?

1) ___________________

2) ___________________

3) ___________________

PRACTICE VARIABLE ON BOTH SIDES: Combine Like Terms First

1) 10p – 7p + 5 = 5p + 21

2) 46 + 6x = 2 – 3x – 2x

3) -9s + 3s + 8 = 12 + 5s – 11s

4) 12x – 7x + 9 = 3x + 6

5) 17 + 6r – 5 = 13r + 12 – 7r

6) 11m + 12 + 5m = 47 + 7m – 8

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PRACTICE VARIABLE ON BOTH SIDES: DISTRIBUTIVE PROPERTY

1) 2(5x + 3) = 7x – 8

2) 7(m – 2) = 4(3m + 4)

3) 4r – 17 = 2(r – 8)

4) -2(b – 7) = 14 – 2b

5) 3x + 12 = 6(3 – x)

6) -2(7 – 3s) = 4s+ 2

7) 3(2y + 4) = 4(y + 5)

8) 4(2p + 9) = 8(5+ p)

PRACTICE VARIABLE ON BOTH SIDES: SIMPLIFYING EXPRESSIONS

1) 3(2x – 7) = 5x – x – 11

2) -2x + 15 + 13x = 10 – 5(7 – x)

3) 6x + 7(x + 2) = 23 + 13x –9

4) 13 – 9(2x + 3) = -4(5x + 4) + 2x

5) 2(9 + 4x) + 6(2 – x) = 7x

6) 6(2x + 5) – 2(7x + 8) = 4(x + 8)

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