Core Curriculum

Pre Algebra Lessons

Secondary


 
Algebraic Integers
Algebra and Rational Numbers
Algebra and Data Analysis
Linear Equations and Inequalities
Functions
Introduction to Functions...So far, the types of linear equations we explored were all in one variable. We have dealt with equations such as 3x + 5 = 11 or (9x – 12) + 23 = -4(x + 2) which at most can have only one solution. Even in some equations we do not reach the same unique solution. Now we are going to deal with the same type of equations, that is, linear equation, but with two variables. In such cases, the answer of an equation is not limited to one; there can be an infinite number of answers for both variables. Linear equations in two variables are the most common mathematical models we use for applied purposes. Their structures are well-fit with many real data with linear relationships. In a linear relationship, two quantities are associated in which one variable is equal to the product of the other variable and a constant number plus a fixed value. The constant number is called the slope of the equation, and represents the rate of change in variables. For example, if y = -900x + 14000 represents the value of a car x years after purchase, it is the constant number or slope -900 that indicates how the car depreciated as time passed. Having such a model, such as the depreciation of a car, we can make reliable predictions about the value in the future. This can be done by drawing the graph of the function and finding its value in the upcoming years from the graph. The fixed value is the value of the car at the time of purchase; that is, when x = 0. This is called the y-intercept of the function. So, the y-intercept of this function is 14000.
Systems of Equations...When solving one linear equation only, the task involved simple steps to reach the solutions. Here we will be dealing with a higher form of math: two equations in two variables. Pairs of equations might not be cooperative when we are struggling to find the unique solution. Sometimes they just do not work together! When solving two equations, if both of their left sides are the same, but their right sides are different, they are called inconsistent equations. If after simplification of one of the equations, you reach the other equations, then the system is called dependent. Think back to one linear equation containing two variables. Often, we can find an infinite number of solutions for x and y. For example, let the equation L + W = 45 represent the relationship between a length (L) and a width (W) of a rectangle. There are an infinite number of rectangles whose dimensions fit in this equation. But, if we are asked to find a certain rectangle whose dimensions not only fit in this equation but the ratio of its dimensions, for example 1:15. Solving this system yields only one solution for L and one solution for W. That is, the solution of this system will choose one out of thousands of the solutions of the first equation, which also satisfies the second equation. If in solving one linear equation, we are looking for an infinite number of solutions. In the case of systems of equations, we look for one solution from each equation that is the same. In other words, we look for a single solution that can be found among the infinite number of solutions of both equations. The difference between an inequality and an equation is only one aspect of algebra. One conveys one unique value; the other may convey an infinite number of values.
Algebra and Geometry
Algebra and Statistics
Polynomials
Introduction to Polynomials...We now turn to another important class of functions; polynomials. We already know that various algebraic operations may be applied to a group of numbers at the same time. For example, in -5(83) * (12 – 47) + 19, the operations addition, subtraction, multiplication, and exponentiation are applied to numbers -5, 8, 3, 12, 4, 7, and 19. If some or all of these numbers are replaced with parameters and variables, they form an algebraic expression. A similar algebraic expression to this numerical one is -4(x3) * (5 – y3) – z, where x, y, and z are variables. We will first study a special class of such expressions in which all the factors are in product form; in other words, all the components creating an expression are multiplied. The expression 4x3y is a routine example of such expressions. All the factors 4, x, and y are multiplied. We call such expressions monomials. Polynomials are simply the sum of a group monomials. When solving problems, more often you will come across combinations of many polynomials that need to be simplified. To do so, we will use addition and subtraction operations using the general properties of real numbers. Knowing the different operations of polynomials and various methods of manipulating polynomials are vital to solving many real life situations. From modeling the cost and revenue of a manufacturer or modeling the population of human being or a certain species in a community, a common mathematical concept is polynomials. In such real life phenomena, the rate of change is not a fixed value. The cost of producing a certain part may be reduced by increasing the number of parts or the population of a certain species in a habitat may be decreased as its size extended during a scientific investigation. All such phenomena can be described by polynomials. Using polynomials as a basic tool in events in which the rate of change is not a fixed value helps us to have relatively precise predication of variations and outcomes of such phenomena in the future.
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