# Logic and Proof

# Algebraic Proofs Exercises

### Example 1

If 4*x* + 12 = 0, then prove that *x*^{2} + 2 = 11 using an algebraic proof.

Statements | Reasons |

1. 4x + 12 = 0 | ? |

### Example 2

If 4*x* + 12 = 0, then prove that *x*^{2} + 2 = 11 using an algebraic proof.

Statements | Reasons |

1. 4x + 12 = 0 | Given |

2. 4x = -12 | ? |

### Example 3

If 4*x* + 12 = 0, then prove that *x*^{2} + 2 = 11 using an algebraic proof.

Statements | Reasons |

1. 4x + 12 = 0 | Given |

2. 4x = -12 | Subtract 12 from (1) |

3. x = -3 | ? |

### Example 4

If 4*x* + 12 = 0, then prove that *x*^{2} + 2 = 11 using an algebraic proof.

Statements | Reasons |

1. 4x + 12 = 0 | Given |

2. 4x = -12 | Subtract 12 from (1) |

3. x = -3 | Divide (2) by 4 |

4. x^{2} = 9 | ? |

### Example 5

If 4*x* + 12 = 0, then prove that *x*^{2} + 2 = 11 using an algebraic proof.

Statements | Reasons |

1. 4x + 12 = 0 | Given |

2. 4x = -12 | Subtract 12 from (1) |

3. x = -3 | Divide (2) by 4 |

4. x^{2} = 9 | Square (3) |

5. x^{2} + 2 = 11 | ? |

### Example 6

Give the statements and the reasons that prove that if 5 – *x*^{2} = 1 and , then *y* = 28.

Statements | Reasons |

1. 5 – x^{2} = 1 | Given |

2. | Given |

3. ? | ? |

### Example 7

Give the statements and the reasons that prove that if 5 – *x*^{2} = 1 and , then *y* = 28.

Statements | Reasons |

1. 5 – x^{2} = 1 | Given |

2. | Given |

3. -x^{2} = -4 | Subtract 5 from both sides of (1) |

4. ? | ? |

### Example 8

Give the statements and the reasons that prove that if 5 – *x*^{2} = 1 and , then *y* = 28.

Statements | Reasons |

1. 5 – x^{2} = 1 | Given |

2. | Given |

3. -x^{2} = -4 | Subtract 5 from both sides of (1) |

4. x^{2} = 4 | Multiply (3) by -1 |

5. ? | ? |

### Example 9

Give the statements and the reasons that prove that if 5 – *x*^{2} = 1 and , then *y* = 28.

Statements | Reasons |

1. 5 – x^{2} = 1 | Given |

2. | Given |

3. -x^{2} = -4 | Subtract 5 from both sides of (1) |

4. x^{2} = 4 | Multiply (3) by -1 |

5. | Substitute (4) into (2) |

6. ? | ? |