# Linear Equations in One Variable

An **equation** is a sentence that expresses the equality of two algebraic
expressions. Consider the equation

2x + 1 = 7.

Because 2(3) + 1 = 7 is true, we say that 3 **satisfies** the equation. No
other number in place of x will make the statement 2x + 1 = 7 true. However, an
equation might be satisfied by more than one number. For example, both 3 and -3
satisfy x^{2} = 9. Any number that satisfies an equation is called a **
solution** or root to the equation.

**Solution Set**

The set of all solutions to an equation is called the **solution set** to
the equation.

The solution set to 2x + 1 = 7 is {3}. To determine whether a number is in the solution set to an equation, we simply replace the variable by the number and see whether the equation is correct.

**Example **

**Satisfying an equation**

Determine whether each equation is satisfied by the number following the equation.

a) 3x + 7 = -8, -5

b) 2(x - 1) = 2x + 3, 4

**Solution**

a) Replace x by -5 and evaluate each side of the equation.

3x + 7 | = -8 | |

3(-5) + 7 | = -8 | |

-15 + 7 | = -8 | |

-8 | = -8 | Correct |

Because -5 satisfies the equation, -5 is in the solution set to the equation.

b) Replace x by 4 and evaluate each side of the equation

2(x - 1) | = 2x + 3 | |

2(4 - 1) | = 2(4) + 3 | Replace x by 4. |

2(3) | = 8 + 3 | |

6 | = 11 | Incorrect |

The two sides of the equation have different values when x = 4. So 4 is not in the solution set to the equation.