# Math 3: Algebra 2

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

Content:• Intersections

• Ways of writing linear equations

• Equation systems

• Systems of inequalities

• Functions

## Intersections

The intersections are those points on the line that intersect with the X or Y axes. The point that intersects the Y line with the X axis is called the * abscissa at the origin*; while the point that intersects the line with the Y axis is called the

**.**

*ordinate to the origin*## Ways of writing linear equations

Linear equations have various forms in which they are presented, each having its own name and utility.

The first is called the **slope - ordinate to the origin form**, and it is the one in which m the value of the slope and b is the value of the origin-ordinate within the equation in its form y = mx + b. For example, the equation Y = 2x + 3, which can be correctly graphed knowing that its slope is equal to 2 and the ordinate to the origin (Y intersection) is equal to 3.

The second form is the **point - slope form**, and it is the one in which they use the coordinates of any point within the line and the slope to construct an equation, which is placed in the form Y - b = m (x - a) where m is the slope and (a, b) are the values of x, y respectively. For example, in a line where m = 2 and that passes through the point (-7,5). The equation can be written without the need to graph in the way y - 5 = 2 [x - (-7)].

The point-slope form can be converted to its slope - ordinate to the origin form through a clearance. For example y - 5 = 2 (x +7) which is solved for y = 2x +19.

The **general form** or **standard form** is that in which the equation is presented in the form Ax + By = C in which A and B are constant in relation to C for the line. To graph this type of equation, a table in T is used, which establishes the value of Y for each given value of X. When we are presented with an equation in its general form and we are asked to obtain the slope, or the ordinate / abscissa at the origin, we use the table T with values at 0 for x or y so that the interection is obtained, and for the slope we use the formula m = y_{2} - y_{1} / x_{2} - x_{1} without the need for a graph. For example with the ecuation 2x + 5y = 50.

X | Y |
---|---|

0 | 2(0) + 5y = 50 → 5y = 50 → y = 10 |

2x + 5(0) = 50 → 2x = 50 → x = 25 | 0 |

To obtain the slope in this example we use the formula.

m = 10 - 0/ 0 - 25

m = 10/ -25**m = -0.4**

Once you have these data, it is easy to write the equation in its **slope - ordinate to the origin form form**. Which would be Y = -0.4x + 10.

To write the equation in its **point-slope form**, the values of any point on the line are taken and they are used together with the slope, in this case we take any value for x / y and take its coordinate on the missing axis. For example, if we take a value of 2 for y and substitute it into the equation

2x + 5(2) = 50

2x + 10 = 50

2x = 40

x = 40/2**x= 20**

So we take the coordinates of the point (20, 2) and the slope -0.4 and write the equation:

y - 2 = -0.4 (x - 20)

To go from **slope forms - ordinate** to the **origin form or point-slope** to the **general form** we need to solve it:

y - 6 = -2/3(x + 3)

y = -2/3x - 2 + 6

y = -2/3x + 4 (**origin form or point-slope form**)

2/3x +y = 4

3(2/3x +y = 4)

2x + 3y = 12 (**general form**)

## Equation systems

A system of equations is one in which we have two equations related to the same problem and that have the same variables, so that in both X has the same value and Y also. These systems of equations are commonly used in verbal problems and have various ways of being solved.

**Graphic method:** Through this, the graphing of two lines is used, one for each equation, and they are used to determine an intersection between the two lines as the answer coordinates to the problem. For example:

x + y = 900

5x + 10y = 5500

X | Y |
---|---|

0 | 0 + y = 900 → y = 900 |

x + 0 = 900 → x = 900 | 0 |

X | Y |
---|---|

0 | 5(0) + 10y = 5500 → 10y = 5500 → y = 550 |

5x + 10(0) = 5500 → 5x = 5500 → x = 1100 | 0 |

**Substitution method:** Through this one of the two unknowns is cleared as a function of the other in one of the two equations. Then the value obtained is substituted on the other equation. For example:

x + y = 6

x - y = 4

1 ) y = 6 - x

2 ) x - (6 - x) = 4

2x - 6 = 4

2x = 10__x = 5__

3 ) 5 + y = 6

y = 6 - 5__y = 1__

**Reduction method:** With this method an unknown is eliminated seeking to make it equivalent in both equations but with opposite coefficients. For example:

x + 2y = 25

2x + 3y = 40

1 ) -2 (x + 2y = 25)

*-2x* - 4y = -50*2x* + 3y = 40

0x - y = -10

2 ) __y = 10__

x + 2(10) = 25

x + 20 = 25

x = 25 - 20__x = 5__

**Equalization method:** In this method one of the two variables in both equations is cleared so that we only have one single variable. Then both are put into a single equation. For example:

2x - y = -1

3x + y = 11

1 ) -y = -1 - 2x

y = 1 + 2x

2 ) y = 11 - 3x

3 ) 11 - 3x = 1 + 2x

11 - 1 = 2x + 3x

10 = 5x__x = 2__

4 ) 2(2) - y = -1

4 - y = -1

-y = -1 - 4

-y = -5__y = 5__

## Systems of inequalities

Inequalities, even when they are not answered with a value but with a set of values, can be graphed. For example:

y ≤ 5/3x - 25/3

m = 5/3

b = - 25/3

As with equations, inequalities can form systems in which a solution set is sought for two inequalities. These sets can only be obtained from graphing. For example:

y > x - 8

y < 5 - x

m_{1} = 1

m_{2} = -1

b_{1} = -8

b_{2} = 5

## Functions

Functions are those terms that determine what will happen to the variable to which they are applied by means of a statement in the form of an equation. For example:

f (x) = x^{2} + 3

f(1) = 1^{2} + 3

f(1) = 4

The function f (x) in this case determines that any value given to x will obey x^{2} + 3.

In summary: a function is a mathematical model that relates two variables, these variables are known as a set. In these sets, each element of the first set (x) is related to a single element (y) of the second set.

The first set is called the **domain** of the function. The values of this represent the independent variable of the function.

The output value for each element of f (x) is called the **image** and the set of images, known as the **path**, represents the dependent variable.

Seeing it graphically within the example function f (x) = x^{2} where x = 1.

- The image is equal to 1, since we replace the value x for 1. [f(1) = 1
^{2}= 1] - The domain is the set of real numbers since no matter wich number you replace x for, the square of any real number would be equal to another real number.
- The path is y ≥ 0 since the square of any real number, positive or negative would be equal to another real number always positive, and the square of 0 is 0.

Functions can be used to represent some real-life situations in the form of mathematical models. For example:

"A rectangular piece of cardboard measures 30 cm long by 15 cm high, in each of the corners squares of x cm long are cut. This cardboard is folded to form a box L cm long and A cm wide."

a ) Write an expression in terms of x for L and A.

b ) Find a function for the volume, V as a function of x.

a ) __L = 30 - 2x____ A = 15 - 2x__

b ) V = (30 - 2x)(15 - 2x)(x)

V = (30x - 2x^{2})(15x - 2x^{2})

V = 450x^{2 }- 60x^{3} - 30x^{3} + 4x^{4}__V = 4x__^{4}__ - 90x__^{3}__ + 450x__^{2}

**© 2021 Daniela Alejandra Rodríguez Cerda**

## Comments

**Umesh Chandra Bhatt** from Kharghar, Navi Mumbai, India on April 13, 2021:

Very nice.