Algebra Articles

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.

Article / Updated 12-21-2022

So, what is an exponent anyway? According to the Oxford dictionary, an exponent is defined as "a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression." Exponents are used in almost all levels of math, from algebra to calculus to physics. Here are two ways you can work with exponents when they show up in formulas and equations. How to multiply exponents You can multiply many exponential expressions together without having to change their form into the big or small numbers they represent. When multiplying exponents, the only requirement is that the bases of the exponential expressions have to be the same. So, you can multiply because the bases are not the same (although the exponents are). To multiply powers of the same base, add the exponents together: If there’s more than one base in an expression with powers, you can combine the numbers with the same bases, find the values, and then write them all together. For example, Here's an example with a number that has no exponent showing: When there’s no exponent showing, such as with y, you assume that the exponent is 1, so in the above example, you write How to divide exponents You can divide exponential expressions, leaving the answers as exponential expressions, as long as the bases are the same. To divide exponents (or powers) with the same base, subtract the exponents. Division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base. For example, Pretty easy, huh? Now wrap your brain around this: Any number to the power of zero equals 1, as long as the base number is not 0.

Article / Updated 08-17-2022

Scientific notation is a standard way of writing very large and very small numbers so that they're easier to both compare and use in computations. To write in scientific notation, follow the form where N is a number between 1 and 10, but not 10 itself, and a is an integer (positive or negative number). You move the decimal point of a number until the new form is a number from 1 up to 10 (N), and then record the exponent (a) as the number of places the decimal point was moved. Whether the power of 10 is positive or negative depends on whether you move the decimal to the right or to the left. Moving the decimal to the right makes the exponent negative; moving it to the left gives you a positive exponent. To see an exponent that's positive, write 312,000,000,000 in scientific notation: Move the decimal place to the left to create a new number from 1 up to 10. Where's the decimal point in 312,000,000,000? Because it's a whole number, the decimal point is understood to be at the end of the number: 312,000,000,000. So, N = 3.12. Determine the exponent, which is the number of times you moved the decimal. In this example, you moved the decimal 11 times; also, because you moved the decimal to the left, the exponent is positive. Therefore, a = 11, and so you get Put the number in the correct form for scientific notation To see an exponent that's negative, write .00000031 in scientific notation. Move the decimal place to the right to create a new number from 1 up to 10. So, N = 3.1. Determine the exponent, which is the number of times you moved the decimal. In this example, you moved the decimal 7 times; also, because you moved the decimal to the right, the exponent is negative. Therefore, a = –7, and so you get Put the number in the correct form for scientific notation When you get used to writing numbers in scientific notation, you can do it all in one step. Here are a few examples: Order of magnitude Why does scientific notation always use a decimal between 1 and 10? The answer has to do with order of magnitude, which is a simple way to keep track of roughly how large a number is so you can compare numbers more easily. The order of magnitude of a number is its exponent in scientific notation. For example, 703 = 7.03 x 102 — order of magnitude is 2 600,000 = 6 x 105 — order of magnitude is 5 0.00095 = 9.5 x 10–4 — order of magnitude is –4 Every number between 10 and 100 has an order of magnitude of 1. Every number between 100 and 1,000 has an order of magnitude of 2.

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).

Cheat Sheet / Updated 04-19-2022

The best way to figure out how the different algebraic rules work and interact with one another is to practice with lots of problems. And Algebra II requires lots of practice. So be prepared to solve equations and systems, graph lines, tackle functions, and so much more.

Cheat Sheet / Updated 04-12-2022

Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.