# Expressions and variables

An algebraic expression comprises both numbers and variables together with at least one arithmetic operation.

**Example**

$$4\cdot x-3$$

A variable, as we learned in pre-algebra, is a letter that represents unspecified numbers. One may use a variable in the same manner as all other numerals:

Addition | 4+y | 4 plus y |

Subtraction | x-5 | x minus 5 |

8-a | 8 minus a | |

Division | z/7 | z divided by 7 |

14/x | 14 divided by x | |

Multiplication | 9x | 9 times x |

To evaluate an algebraic expression you have to substitute each variable with a number and perform the operations included.

**Example**

Evaluate the expression when x=5

$$4\cdot x-3$$

First we substitute x with 5

$$4\cdot 5-3$$

And then we calculate the answer

$$20-3=17$$

An expression that represents repeated multiplication of the same factor is called a power e.g.

$$5\cdot 5\cdot 5=125$$

A power can also be written as

$$5^3=125$$

Where 5 is called the base and 3 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.

$$5^3=5\cdot 5\cdot 5$$

$$3^1$$ | 3 to the first power | $$3$$ |

$$4^2$$ | 4 to the second power or 4 squared | $$4 \cdot 4$$ |

$$5^3$$ | 5 to the third power or 5 cubed | $$5\cdot5\cdot5$$ |

$$2^6$$ | 2 to the sixth power | $$2\cdot2\cdot2\cdot2\cdot2\cdot2$$ |

Evaluate the following expression when \( x = 2 \) and \( y = -3\),

\[ x^2 - y + 2x \]

\[ \begin{align} (2)^2 - (-3) + 2(2) &= 4 + 3 + 4 \\ &= 11. \end{align} \]

\[ 10x + yx - y^2 \]

\[ \begin{align} 10(2) + (2)(-3) - (-3)^2 &= 20 - 6 - 9 \\ &= 20 - 15 \\ &= 5. \end{align} \]

**Video lesson**

Evaluate the expression when x=4 and y=3

\( 5x + y^{2}- xy \)