Math Skills ReviewAlgebraic Manipulation |

The key to solving simple algebraic equations containing a single unknown (e.g. x + 6 = 10) is to realize that the equation is an equality. As long as you do the same mathematical operation (e.g. add a constant, subtract a constant, multiply by a constant, and divide by a constant) to both sides of the equation, the equality is still an equality. This includes squaring both sides of the equation or taking the square root of both sides of the equation.

**Fundamental Laws:**

- Distributive Law: 3(x + 2) = 3x + (3)(2) = 3x + 6
- Associative Law: 4x - 7x = x(4 - 7) = -3x

Example 1 | To solve for x, it is necessary to subtract 6 from both sides of the equation | |

Example 2 | To solve for x, you need to add 6 to both sides of the equation and then divide both sides by 2. | |

Example 3 | To isolate x, you need to (1) multiply through by 6, (2) subtract 2 from both sides, and (3) divide both sides by 5. | |

Example 4 | To solve for x this time, you need to (1) multiply both sides of the equation by 4 and 3 to cancel out the denominator in line 2, | |

Example 4(alternative) | This process is called "cross-multiplying." This entails multiplying the numerator of one side of the equality by the denominator of the other side of the equality. When this is done, the very same line 3 results. The rest of the problem is done identically. | |

Example 5 | This problem could be very complicated and become a quadratic equation. However, because it has a perfect square on both sides, if you simply take the square root of both sides of the equality, you are left in line 3 with a straightforward algebra problem as you solve for the positive root, which I did here.
In Chemistry, when we use this technique to solve equilibrium problems, only one of the roots is meaningful. Of course, the square root of 49 can be -7 as well as +7. You can then go ahead and solve for the second root, x = -0.8 = -4/5. |

**QUIZ: Solve for x.**

Question 1 | Question 2 | Question 3 | Question 4 |

Dimensional Analysis | Significant Figures | Manipulation of Exponents |

Scientific Notation | Logarithms | The Quadratic Equation |