Introduction to Algebra

What Is Algebra? The word, Algebra is an Arabic word meaning bringing together broken parts. This useful tool of mathematics was invented by the 9th century Arab mathematician, Mohammed Ibn Musa Al-Khwarizmi (from whos Europeanised name comes the word, algorithm). Algebra allows the finding of unknown numbers from information given. Here are two examples of the type of problems that can be solved using algebra. Example 1: When six is added to this number, it gives nine. What is the number? Example 2: A number is multiplied by itself. Three times that same number is subtracted. Finally, two is added. The total is now zero. What is the number? Example 1 is trivial. The number has to be three. Example 2 is much harder to guess. I will return to it later. Rules And Notation In algebra, letters are used in place of numbers that are not known. These letters are then manipulated in accordance with certain rules until an answer appears. The usual letter for the unknown number is X. Normally, small x is used but this is too similar to the multiplication sign (×) so I will use capital X. Two numbers added together are shown as: A + B Two numbers subtracted are shown as: A -B Two numbers multiplied together are shown as: A × B but more commonly: AB Two numbers divided are shown as: A /B A number multiplying a sum of numbers is shown as: A(B + C) This can be expanded by multiplying everything inside the brackets by the number outside: AB + AC A number dividing a sum of numbers is shown as: (B + C) /A This can be expanded by dividing everything inside the brackets by the number outside: B /A + C /A Two positive numbers multiplied together give a positive number: A × A = B Two negative numbers multiplied together also give a positive number: -A × -A = B A negative number multiplied by a positive number gives a negative number: -A × A = -B Solving Simple Equations Returning to Example 1 from above, this can be written in algebra as X + 6 = 9 This is called an equation because there is an equals sign. If we want to find the value of the unknown number, X, we have to get the X on one side of the equation and the numbers on the other side. We can do whatever we like to this equation as long as we do the same to both sides of the equation. We can easily isolate the X by subtracting 6 from both sides: X + 6 = 9 X + 6 -6 = 9 -6 X = 3 So the unknown number is 3. It is always a good idea to check an answer by puting the value 3 back in the original equation: Checking by replacing X with 3 gives: 3 + 6 = 9 ..... correct. Example 2 from above will be dealt with later. Example 3: Solve the equation X -6 = 2. Again, we have to isolate the X on one side of the equation and the numbers on the other side. This can be done by adding 6 to both sides. X -6 = 2 X -6 + 6 = 2 + 6 X = 2 + 6 X = 8 Checking by replacing X with 8 gives: 8 -6 = 2 ..... correct. Example 4: Solve the equation 2X -3 = -1 This one is a little more complicated. The X is multiplied by 2 and then 3 is subtracted. We must deal with the 3 being subtracted first. We can do this by adding 3 to both sides. 2X -3 = -1 2X -3 + 3 = -1 + 3 2X = 2 The next step is to divide both sides by 2. 2X = 2 2X /2 = 2 /2 X = 1 Checking by replacing X with 1 gives: (2 × 1) -3 = 2 -3 = -1 ..... correct. Example 5: Solve the equation, 2(X + 3) = 4. In this equation, the 2 multiplies everything inside the bracket. This 2 must be dealt with first. It is removed by dividing both sides by 2. 2(X + 3) = 4 2(X + 3) /2 = 4 /2 (X + 3) = 2 X + 3 = 2 Now we can subtract 3 from both sides. X + 3 -3 = 2 -3 X = -1 Checking by replacing X with -1 gives: 2 × (-1 + 3) = 2 × 2 = 4 ..... correct. Example 6: Solve the equation 4 /(X + 1) = 2. In this equation the part containing the X is at the bottom. The X + 1 in the bracket is all dividing the 4. It must be brought to the top. This is done by multiplying both sides by (X + 1). 4 /(X + 1) = 2 4 = 2(X + 1) The next step is to divide both sides by 2. 4 /2 = X + 1 2 = X + 1 Subtract 1 from both sides. 2 = X + 1 2 -1= X + 1 -1 1= X or X = 1 Checking by replacing X with 1 gives: 4 /(1 + 1) = 4 /2 = 2 ..... correct. Example 7: Solve the equation -X -3 = -2. The X has a minus sign in front of it. It is being multipied by -1: -1X -3 = -2 To remove the -1 we need to multiply both sides by -1. -1X -3 = -2 -1 × (-1X -3) = -2 × -1 X + 3 = 2 Multiplying by -1 is the same as changing the signs throughout. The next step is to subtract 3 from both sides. X + 3 -3 = 2 -3 X = -1 Checking by replacing X with -1 gives: -(-1) -3 = 1 -3 = -2 ..... correct. Note that two minuses multiplied together make a plus. Example 8: Solve the following equation, 2X -5 = 4 -X. The usual rules apply: get the X on one side and the numbers on the other side. Add X to both sides to remove the X from the right hand side. 2X -5 = 4 -X 2X -5 + X = 4 -X + X 2X + X -5 = 4 Two Xs and a single X can be added to give 3 Xs. 2X + X -5= 4 3X -5 = 4 Add 5 to both sides. 3X -5 = 4 3X -5 + 5 = 4 + 5 3X = 9 Divide both sides by 3. 3X = 9 3X /3 = 9 /3 X = 3 Checking by replacing X (in 2X -5) with 3 gives: (2 × 3) -5 = 6 -5 = 1 Checking by replacing X (in 4 -X) with 3 gives: 4 -3 = 1 ..... both sides agree.