• A *linear expression* in a variable *x* is one that simplifies to *ax + b*, where* a* and *b* are numbers.

A *linear equation* in x says that two linear expressions are equal.

A linear equation can be simplified by collecting terms on each side of the equation.

At worst, the result will be an equation such as 5x + 5 = 2x + 29, with a variable term and a constant on each side of the equals sign.

•This sort of equation is easy to solve, as shown in the chart below.

The basic idea is that you want both sides of the equation to stay equal, and so you should always "do the same thing" to both sides of the equation. In detail, you can:

- Add any number to both sides of the equation;
- Subtract any number from both sides;
- Multiply or divide both sides by any number other than zero.

Here is the original equation. On the left side, the constant 5 has been added to 5x. | 5x + 5 = 2x + 29 |

Subtracting 5 from both sides makes the left side simpler. | 5x + 5 -5 = 2x + 29 -5 |

Simplify. | 5x = 2x + 24 |

Subtract 2x from both sides to get all variable terms on the left side. | 5x - 2x = 2x + 24 -2x |

Simplify | 3x = 24 |

Divide both sides by 3 and simplify to get the solution. | x = 24/3 so x = 8 |

• When you get more experience, and if it's OK with your teacher, condense the solution chart as follows.

Here is the original equation. | 5x + 5 = 2x + 29 |

Subtract the left side constant from both sides and simplify. | 5x = 2x + 24 |

Subtract the right side variable term from both sides and simplify. | 3x =24 |

Divide by 3 and simplify | x = 8 |

• One way to think about the method shown is to say that at each step, we t*ransposed a term from one sided to the other. *

•*Transpose* means:* remove a term from one side* of the equation and *insert in on the other side with its sign reversed.*

This is just another way of saying that you reverse adding a term by subtracting that term. Let's rewrite the solution based on that idea.

Here is the original equation. On the left side, the term 5 is added to 5 | 5x + 5 = 2x + 29 |

Transpose the term 5 from the left side to the right side. | 5x = 2x + 29 - 5 |

Simplify. On the right side, the term 2x is added to 24 | 5x = 2x + 24 |

Transpose 2x from the right side to the left side | 5x - 2x = 24 |

Simplify | 3x = 24 |

Divide both sides by 3 and simplify to get the solution. | x = 24/3 = 8 |

•Now look at the figure.. The problem shown is the one solved above. You can try any of three methods:

- Copy the steps of the solution shown, mainly to get practice with typing.
- Solve the problem with the steps visible as hints in the left column.
- Solve the problem without hints. Warning: you still have to follow the (invisible) hints line by line!

•To check your work, click the 'Check my work' button at any time. You will be told whether each of the lines you typed is correct. If a line says 'No', retype your answer.

•To get a new problem, click on the top button in the list.