Mary Jane Sterling

Article / Updated 05-30-2024

A matrix is a rectangular array of numbers. Each row has the same number of elements, and each column has the same number of elements. Matrices can be classified as: square, identity, zero, column, and so on. Where did matrices come from? For most of their history, they were called arrays. There are references to arrays in Chinese, French, Italian, and many other mathematical works going back many hundreds of years. American mathematician George Dantzig's work with matrices during World War II allowed for the coordination of shipments of supplies and troops to various locations. Matrices are here to stay. You may be familiar with a method used to solve systems of linear equations using matrices, but this application just scratches the surface of what matrices can do. First, just in case you're not familiar with solving equations using matrices, let me give just a quick description. If you want to solve the following system of equations: You write the matrix: And then you perform row operations until you get the matrix: From that matrix, you know that the solution of the system of equations is x = 1, y = -3, and z = -5. Pretty slick, don't you think? But uses for matrices don't stop there. You can solve traffic control problems, transportation logistics problems (how much of each item to send to various distribution centers), dietary problems (how much of each food product is needed to meet several different dietary requirements), and so on. Matrices work well in graphing calculators and computer spreadsheets — just set up the problem and let the technology do all the work.

Article / Updated 08-28-2023

When you want to count up how many things are in a set, you have quite a few options. When the set contains too many elements to count accurately, you look for some sort of pattern or rule to help out. Here, you practice the multiplication property. If you can do task one in m1 ways, task two in m2 ways, task three in m3 ways, and so on, then you can perform all the tasks in a total of m1 · m2 · m3 . . . ways. Sample questions How many ways can you fly from San Francisco to New York City, stopping in Denver, Chicago, and Buffalo, if the website offers four ways to fly from San Francisco to Denver, six ways to fly from Denver to Chicago, two ways to fly from Chicago to Buffalo, and three ways to fly from Buffalo to New York City? 144. Multiply 4 x 6 x 2 x 3 = 144. This method doesn’t tell you what all the routes are; it just tells you how many are possible so you know when you’ve listed all of them. (Better get to work on that.) How many ways can you write a password if the first symbol has to be a digit from 1 to 9; the second, third, and fourth symbols have to be letters of the English alphabet; and the last symbol has to be from the set {!, @, #, $, %, ^, &, *, +}? 1,423,656. You multiply 9 x 26 x 26 x 26 x 9 = 1,423,656. This system allows a lot of passwords, but most institutions make you use eight or more characters, which makes the number of possibilities even greater. Practice questions If you have to take one class in each subject, how many different course loads can you create if you have a choice of four math classes, three history classes, eight English classes, and five science classes? How many different ice-cream sundaes can you create if you have a choice of five ice-cream flavors, three sauces, and five sprinkled toppings if you choose one of each type? How many different automobiles can you order if you have a choice of six colors, four interiors, two trim options, three warranties, and two types of seats? How many different dinners can you order if you have a choice of 12 appetizers, 8 entrees, 5 potatoes, 6 desserts, and a choice of soup or salad? Following are answers to the practice questions: The answer is 480. Multiply: 4 x 3 x 8 x 5 = 480. The answer is 75. Multiply: 5 x 3 x 5 = 75. The answer is 288. Multiply: 6 x 4 x 2 x 3 x 2 = 288. The answer is 5,760. Multiply: 12 x 8 x 5 x 6 x 2 = 5,760. Don’t forget that soup or salad is two choices for that selection.

Article / Updated 07-05-2023

In algebra, an improper fraction is one where the numerator (the number on the top of the fraction) has a value greater than or equal to the denominator (the number on the bottom of the fraction) — the fraction is top heavy. Improper fractions can be written as mixed numbers or whole numbers — and vice versa. For example, Practice questions Change the mixed number to an improper fraction. Change the improper fraction to a mixed number. Answers and explanations The correct answer is To change a mixed number to an improper fraction, you need to multiply the whole number times the denominator and add the numerator. This result goes in the numerator of a fraction that has the original denominator still in the denominator. So, do the following math: This means that the improper fraction is The correct answer is To change an improper fraction to a mixed number, you need to divide the numerator by the denominator and write the remainder in the numerator of the new fraction. In this example, to change the improper fraction 16/5 to a mixed number, do the following: Think of breaking up the fraction into two pieces: One piece is the whole number 3, and the other is the remainder as a fraction, 1/5.

Cheat Sheet / Updated 06-22-2023

Here it is. You have this All-in-One reference for concepts and formulas occurring in Algebra II. The material here is grouped by general algebraic content to make it easier to find what you need. The formulas have the standard mathematical format with variables appearing as x, y, and z and the constant numbers appearing as letters at the beginning of the alphabet.

Article / Updated 06-05-2023

In algebra, the distributive property is used to perform an operation on each of the terms within a grouping symbol. The following rules show distributing multiplication over addition and distributing multiplication over subtraction: Practice questions –3(x – 11) = ? Answers and explanations The correct answer is –3x + 33. The correct answer is –5.

Cheat Sheet / Updated 02-09-2023

Trigonometry is the study of triangles, which contain angles, of course. Get to know some special rules for angles and various other important functions, definitions, and translations. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if you plan to study trig for very long.

Article / Updated 08-11-2022

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of tangent, so that all the functions match, making it easier to solve the equation. To rewrite the sine function in terms of tangent, follow these steps: Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left. Replace cosine with its reciprocal function. Solve the Pythagorean identity tan2θ + 1 = sec2θ for secant. Replace the secant in the sine equation.

Article / Updated 08-10-2022

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of cotangent, so that all the functions match, making it easier to solve the equation. To write the sine function in terms of cotangent, follow these steps: Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left. Replace cosine with its reciprocal function. Solve the Pythagorean identity tan2θ + 1 = sec2θ for secant. Replace the secant in the sine equation. Replace all the tangents with 1 over the reciprocal for tangent (which is cotangent) and simplify the expression. The result is a complex fraction — it has fractions in both the numerator and denominator — so it’ll look a lot better if you simplify it. Rewrite the part under the radical as a single fraction and simplify it by taking the square root of each part. Multiply the numerator by the reciprocal of the denominator. Voilà — you have sine in terms of cotangent.

Article / Updated 08-08-2022

The 30-60-90 triangle is shaped like half of an equilateral triangle, cut straight down the middle along its altitude. It has angles of 30 degrees, 60 degrees, and 90 degrees, thus, its name! In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3. The hypotenuse is the longest side in a right triangle, which is different from the long leg. The long leg is the leg opposite the 60-degree angle. Two of the most common right triangles are 30-60-90 and the 45-45-90-degree triangles. All 30-60-90 triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in radians, it translates to the following: The figure illustrates the ratio of the sides for the 30-60-90-degree triangle. If you know one side of a 30-60-90 triangle, you can find the other two by using shortcuts. Here are the three situations you come across when doing these calculations: Type 1: You know the short leg (the side across from the 30-degree angle). Double its length to find the hypotenuse. You can multiply the short side by the square root of 3 to find the long leg. Type 2: You know the hypotenuse. Divide the hypotenuse by 2 to find the short side. Multiply this answer by the square root of 3 to find the long leg. Type 3: You know the long leg (the side across from the 60-degree angle). Divide this side by the square root of 3 to find the short side. Double that figure to find the hypotenuse. In the triangle TRI in this figure, the hypotenuse is 14 inches long; how long are the other sides? Because you have the hypotenuse TR = 14, you can divide by 2 to get the short side: RI = 7. Now you multiply this length by the square root of 3 to get the long side:

Cheat Sheet / Updated 04-28-2022

Formulas, patterns, and procedures used for simplifying expressions and solving equations are basic to algebra. Use the equations, shortcuts, and formulas you find for quick reference. This Cheat Sheet offers basic information and short explanations (and some words of advice on traps to avoid).