| About Us Topics: Introduction Math: Algebra Math in General Language: English Foreign History Biology Physics Humanities | The
process of learning can be frustrating for some of us while for others
it can be fun and easy. Some know instinctively how to learn. Others
of us need to be given the keys to the kingdom of academia. Those keys
are the spelling out of specific steps and techniques for acquiring
mastery of the various disciplines. Even after we're through with formal education we might want to learn a new skill like computer programming or playing a banjo. If you enjoy learning and have little difficulty in mastering a new knowledge set you'll be wasting your time reading further. This site is specifically for those who need some help. We sell no products on this site and make no recommendations on books or anything else to buy. If advertising appears on this website we endorse nothing. Although there are common denominators in every learning experience there are also some differences. We'll start with a skill set that so many students seem to fear - Algebra. Algebra is an extremely simple form of math that requires very little learning. You must learn the few rules. Once those rules are ingested and absorbed the problems seem to work themselves out with very little effort on your part. One of the problems so many students have other than fear is trying to guess the answer. Forget that. Approach the problem with a clear head and work it out one step at a time. Following is a very basic word problem which we'll use algebra to solve. Don't try to solve it by the process of elimination - guessing and testing your guess. Here it is: One clear Spring day two shepherds approached each other in a field as their sheep followed behind. One shepherd say to the other, "If you'll just give me one of your sheep then we'll both have the same number." The other shepherd answered, "Forget that. You give me one of yours and I'll have twice as many as you." How many sheep did each shepherd have? All we're trying to find out is how many sheep each shepherd had as they approached each other. This is called solving for two unknowns. Let x = the number of sheep the first shepherd had. Let y = the number of sheep the second shepherd had. The use of the x and y is arbitrary. We could just as easily use a and b or any other convenient symbols. Now lets solve the problem: We know that x + 1 = y - 1. That's what the problem said. "You give me one of yours and we'll both have the same number." We also know that y + 1 = 2(x-1). That's also stated in the word problem. Now we'll take the x + 1 = y - 1 and get the x by itself by subtracting 1 from each side of the equation. That's one of the rules. Treat both sides of an equation equally. Now we have x = y -2, but we still don't know what y is. We know that x = y-2; so, we're going to plug that value for x into the second equation: y + 1 = 2(x-1). Note that the x-1 is enclosed in parentheses. That's because it's one number, and whatever is done to any part of it must be done to the rest of it. We'll first do the multiplication in the right side of the equation. That gives us y + 1 = 2x - 2. Substituting y - 2 for the x we get y + 1 = 2(y-2) - 2. We do the multiplication which gives us y + 1 = 2y -4 -2. We subtract a y from each side to get 1 = y - 4 -2 = y - 6. Now all we do is add 6 to each side to get the y by itself, and we have 7 = y or y = 7. We know that x = y - 2; so, x = 5. The first shepherd had 5 sheep and the second one had 7. In Algebra never jump the gun and try to guess steps. You just take it one simple step at a time. Lets square the expression (x+3). This is done pretty much the same way as in arithmetic. " x times x = x2 , x times 3 = 3x, 3 times x = 3x, and 3 times 3 = 9." Add it all together and you get x2 + 6x + 9. Now let's test that by giving a value to x. Let x = 6. This time the expression becomes (6 + 3)2. Do the multiplication and you get 36 + 18 + 18 + 9 which equals 81, the same as if you were just squaring 9. This site is not intended to teach you algebra or any other discipline, but to show you the simplicity in learning. What I hope you have learned here in regard to algebra and math in general is that there are rules in arithmetic, algebra, trigonometry, the calculus, plane and solid geometry that you must solidly know. The rules are all simple, but you cannot enjoy or excel in math without learning them so well that they become a part of you. Throw away any thought that the discipline is difficult and have confidence that by taking each problem one step at a time while following the rules your solution will be correct. Check back. There's much more to be added. |