# Math 2: Algebra 1

*A Biomedical science student, certificated in english, spanish, french and japanese language, who also writes diverse topic articles.*

Content• Use of variables

• Dependent and independent variable

• Equations and inequalities

• Linear equations

• Graphics

• Slope of the line

## Use of Variables

In most algebraic expressions it is common to use variables, which are, as their name indicates, are values that vary with respect to the expression.

These variables are commonly represented as letters with no set value.

For example: x + 2 = 5 (this being a simple algebraic expression)

Algebraic expressions can range from the simplest, such as z = 1, or the most complex that include fractions between variables such as x^{2} - 1/2 x + 8.

## Dependent and Independent Variable

Within real life situations and/or algebraic problems derived from them, it can be found that there is more than one variable many times. Within these variables there will always be one defined as the dependent variable and another as the independent variable.

The independent variable is one over which you have control, and that is not modified by other variables. While on the other way, the dependent variable is one over which there is no control, it completely depends on the changes made on the independent variable.

For example, in the hypothetical situation: "High school boy wants to buy a new video game, he set out to help the young son of a friend of his mother with her homework to get money. If the child's mother pays him $ 10 for each day he goes and helps, and his video game costs $ 500. How many days must he go to help with homework to complete his video game?"

In this example we are presented with variables A and B, where A is the number of days that the boy must help the child with his homework, and B is the money he will collect on those days, considering that for each day he gets $ 10. As you can see if the value of A is changed, the value of B will change, which is why it is called a dependent variable.

## Equations and inequalities

Equations and inequalities are those algebraic representations of a situation or problem in real life, easy or complex, often including some formulas used in other sciences such as physics, chemistry, or geometry. An example of a simple equation (known as an equation of the first degree) is x + 50 = 200.

On the other hand, a complex equation can have more than one variable, or a single variable to different powers (wich are known as second, third degree or more), such as x^{3} + 4x^{2} + 2x + 8, or as already mentioned some geometric formula such as the area of a triangle: (b)(a)/2. This ecuations are ussually equal to other numbers or equations in order to be solved.

The inequalities are those that are not equated with other numbers/equations to arrive at a solution, but rather obtain a range of values that would be the possible answer to a variable. For this, the signs <and> are used that mean greater than and less than and that delimit the range of values for the result of the variable. Unlike equations, whose result is represented graphically as a point on a line, with inequalities it is represented as an interval of a line.

## Linear equations

Linear equations are those that represent a specific point on a line on the Cartesian plane. These equations are those of the first degree and have the form Ax + B = 0, in which A must be greater or less than 0 but never equal to 0.

These equations can be found equal to 0 or equal to another equation and are solved by clearing the values to find the unknowns.

For example: 15x + 45 = 20x + 10

15 x + 45 - 20x - 10 = 0

15x - 20x +45 - 10 = 0

-5x + 35 = 0

-5x = -35

x = -35/-5__x = 6__

Equations can have one solution, two, more than two infinite solutions, or no solution.

For example (infinite solutions): -7x + 2 = 2x +2 -9x

-7x + 2 = -7x +2

-7x + 2 +7x -2 = 0

-7x + 7x + 2 - 2 = 0**0 = 0**

For example (no solution): -7x + 3 = 2x +2 -9x

-7x + 3 = -7x + 2

-7x + 3 + 7x - 2 = 0

-7x + 7x + 3 - 2 = 0**1 = 0**

There are linear equations that have unknown coefficients that are expressed with other variables besides X. These equations are usually solved through a common factor and the result is an expression rather than a value.

For example: A(5 - x) = Bx - 8

A5 - Ax = Bx - 8

A5 - Ax - Bx = -8

-1 (-Ax - Bx = -8 - A5)

Ax + Bx = 8 + A5

x(A + B) = 8 + A5**x = 8 + A5/(A + B)**

## Graphics

Linear equations that are expressed by graphs have two variables (x, y), these normally form coordinates that can be located in the Cartesian plane as points on a line that belong to the equation. In these equations, a value in x is normally taken to obtain a value in y successively until a line is formed.

For example:

y = 2x - 3

x | y |
---|---|

-3 | 2(-3) -3 = -9 |

-2 | 2(-2) -3 = -7 |

-1 | 2(-1) -3 = -5 |

0 | 2(0) -3 = -3 |

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

2 | 2(2) -3 = 1 |

3 | 2(3) -3 = 3 |

Linear equations can be written in different ways, so it depends on how the equation is written, the name that it receives, and how it is solved.

## Slope of the Line

The slope is that inclination that a line has in the Cartesian plane. This can also be defined as the relationship between a unit on the X axis with respect to the corresponding units on the Y axis.

If you advance one unit (of the line) along the x axis, you must count how many units from that point you must advance over axis y. In the case that when advancing 1 in x, and from that point, 1 in Y is raised to cross with the line, it would be established that the relationship between both is 1/1.

That is, in this case, the slope would be 1, so that for each unit that is advanced in X, one will also advance in Y.

The slope can be positive or negative, if it is negative, the line will go in the opposite direction to the usual one, that is to say that, speaking of the slope, the relationship would be that for each unit in x, it would advance negative units in Y (down).

One way to calculate the slope of a line without the need to graph it, is through two or more equations in the ** general form** and the formula m = y

_{2}- y

_{1}/ x

_{2}- x

_{1}.

The slope of a completely horizontal or vertical line is always equal to 0, since it does not have a slope as such. Explained through the formula, it would be to say that the change values (x / y2 - x / y1) in X or Y were equal to 0.

**© 2021 Daniela Alejandra Rodríguez Cerda**