Algebra Articles

Article / Updated 09-22-2022

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 08-17-2022

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.

Cheat Sheet / Updated 03-28-2022

Algebra problems are easier to solve when you know the rules and formulas. Unlike other subjects where you can just read or listen and absorb the information sufficiently, math takes practice. The only way to figure out how the different algebraic rules work and interact with one another is to get into the problems and get your hands dirty, so to speak. Be prepared to practice on lots of different problems.

Cheat Sheet / Updated 03-14-2022

To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.

Cheat Sheet / Updated 02-24-2022

Algebra problems are easier to solve when you know the rules and formulas. Memorizing key algebra formulas will speed up your work, too. And if you know the rules of divisibility and the order of operations, you'll be able to solve algebra problems without getting stressed.

Cheat Sheet / Updated 01-21-2022

Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!

Article / Updated 12-21-2021

When the equation of a circle appears in the standard form, it provides you with all you need to know about the circle: its center and radius. With these two bits of information, you can sketch the graph of the circle. The equation x2 + y2 + 6x – 4y – 3 = 0, for example, is the equation of a circle. You can change this equation to the standard form by completing the square for each of the variables. Just follow these steps: Change the order of the terms so that the x's and y's are grouped together and the constant appears on the other side of the equal sign. Leave a space after the groupings for the numbers that you need to add: x2 + 6x _____ + y2 – 4y _____ = 3 Complete the square for each variable, adding the number that creates perfect square trinomials. In the case of the x's, you add 9, and with the y's, you add 4. Don't forget to also add 9 and 4 to the right: x2 + 6x + 9 + y2 – 4y + 4 = 3 + 9 + 4 When it's simplified, you have x2 + 6x + 9 + y2 – 4y + 4 = 16 Factor each perfect square trinomial. The standard form for the equation of this circle is (x + 3)2 + (y – 2)2 = 16. The circle has its center at the point (–3, 2) and has a radius of 4 (the square root of 16). To sketch this circle, you locate the point (–3, 2) and then count 4 units up, down, left, and right; sketch in a circle that includes those points. The figure below shows you the way. With the center, radius, and a compass, you too can sketch this circle.