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In Part 1 of this course (Math 1131), you were newly introduced to the trigonometric functions: sine, cosine, and tangent. You learned how you can use these functions to solve triangles by setting up ratios, but you never learned what they looked like graphed. It turns out that if you…
Many applications in real-life contain two or more unknowns. To solve such problems, we must write as many independent equations as there are unknowns. Otherwise, it is not possible to obtain numerical answers. This section mirrors what you’ve already been doing this unit – that is, solving systems with two or…
There are practically countless methods devised over the past millennia that have enabled mathematicians to solve systems with three unknowns. In this section, we will focus on one method exclusively which is based on the methods that you’ve already learned when you solved systems with two unknowns. Technically, when a…
In this section, you’ll use the method of addition-subtraction or the method of substitution to solve systems of equations with literal coefficients (coefficients that aren’t numbers). To do this, you treat the coefficients as if they were numbers, keeping in mind what you’ve learned in the past about combining like-terms.…
This lesson will explore how to use the techniques used in the previous method to solve non-linear equations. Specifically, we’ll look at a system of equations whose variable is found in the denominator. null Notice that both equations displayed above consist of terms whose variables (x and y) are positioned…
Elimination Method In the elimination method (also known as the addition-subtraction method), we eliminate one of the unknowns by first (if necessary) multiplying each equation by a factor will make the coefficients of one unknown in both equations equal, regardless of their signs. The two equations are then added or subtracted…
This chapter introduces us to solving a system of equations. In part one of this course (Math 1131), you learned how to solve linear equations (such as, 25 = x + 5), for the unknown variables. This time you’ll learn to solve a system of linear equations with two unknowns,…
A numerical operation can be described as an action or process used to solved a numerical problem. Adding and Subtracting Signed Numbers If you have two numbers x and y, the following rules apply when these numbers are being added or subtracted. Rule of Signs for Addition: (x – y)…
To ensure that arithmetic calculations are performed consistently, we must follow the order of operations. If an arithmetic expression contains brackets, exponents, multiplication, division, addition, and subtraction, we use the following procedure: Perform all operations inside a bracket first (the operations inside the bracket must be performed in proper order).…
In mathematics, you will have trouble understanding the material unless you clearly understand the language of the words that are being used. Here are some commonly used words and phrases that you need to know, and that will be repeated throughout the duration of this course. Integers The values: ……
Mean The mean is a number that shows the center of the data. Another word for mean is average and it is mathematically symbolized as x̄ (x-bar). However, when calculating the mean of the whole population, you use the Greek letter μ (mu) instead. To calculate the average of a sample, you sum all the observations,…
The standard deviation (SD) is a measure of the spread of the data (how far from the normal it is). A number such as the mean or the standard deviation may be found either for an entire population (symbolized as σ or σx) or for a sample (symbolized as s) drawn from that…
In statistics, an entire group of people or things is called a population or universe. A population can be infinite or finite. Examples of finite populations include the number of airplanes owned by an airline, or the potential consumers in a target market. Examples of infinite populations include the number of…
A literal equation is one in which some or all of the constants are represented by letters. Arguably any mathematical formula expressing an actual relationship between its variables is a literal equation. Take the Pythagorean theorem formula as an example. null It consists of three variables, a and b are the…
Depending on the number of terms in the equation, these questions can go from being simple to extremely difficult. Easiest of these types is when you have a single term on the left side and a single term on the right. This of course was discussed earlier in this unit…
A key component to multiplying and dividing algebraic fractions is knowing how to do it to ordinary fractions. That being said, you’re first expected to review how it’s done before continuing. If after a few examples you feel confident enough, you may skip it. Multiplying Algebraic Fractions Just as you…
The skills you’ll learn in this lesson will come in handy unexpectedly one day when you’re stuck trying to simplify what appears to be an impossible expression to reduce. Take a look at the three expressions below: null null null At first glance, you might be questioning how do I…
Although this concepts was briefly discussed in one of the earlier lessons, you learned that sometimes you may need to combine several of the techniques to factor a single expression. As a result, to factor the expressions found in this section, you’ll have to device a plan before starting because…
Fractions always have a tendency to scare math student no matter what level of study they’re in. You’re likely to have been first introduced to fractions in elementary school, so before we start mixing in variables into our questions – as you’d expect with algebraic expressions – a good place to…
Generally, the higher the degree of a polynomial, the harder it becomes to factor. The highest degree you’re expected to factor in this course are cubic equation (those raised to the power of three). Specifically, we’ll look at examples similar in structure to quadratics that are a difference of square,…