Class 3 -- Expressions I I. Solving linear equations A. Addition B. Multiplication C. Addition and Multiplication Together II. Commutative, Associative, and Distributive Properties A. Commutative property B. Associative property C. Distributive property III. Factoring out common factors This week's class is devoted to manipulating the algebraic expressions we learned to construct last week. The basic idea of manipulating algebraic expressions is that algebraic expressions describe a rule for doing a particular arithmetic operation or set of operations, and it is possible to express that rule in several equivalent ways. I. Solving Basic Equations The most useful sorts of manipulations are those that let us solve equations. Manipulating expressions isn't very useful unless we can actually figure out what some of the unknowns are. The simplest type of equation to solve is one involving just a single variable multiplied by or added to constants. These are called linear equations. We can solve all linear equations using just two ideas. A. Addition Principle Let's consider a simple english statement: "Some number decreased by eight is thirteen" (on board) What's that in algebra? (have class answer) It's "x - 8 = 13" (on board) We can look at this and just guess the answer, but I want to show you a formal trick to deal with equations that look like this. The neat thing is that, with an equation, I can add or subtract anything I want to one side -- as long as I also do it to the other. What I mean by this is that if I have the equation "1 = 1" (on the board), I can also get a true equation by adding two to both sides: "1 + 2 = 1 + 2" (on board). Adding the same thing to both sides leaves a true equation true. This is called the "addition principle". So, with "x - 8 = 13", suppose I add eight to both sides: (on board) x - 8 = 13 (on board) +8 +8 ---------- x = 21 Notice that the plus eight cancels out the minus eight! I'm left with an equation that is just the variable by itself -- so it's obvious here that x is 21. We can see a principle here: if we want to figure out what the variable is, we want to get the variable by itself on one side of the equation. To do that, we can add or subtract to cancel out whatever is with the variable. Thus, if I had "x + 3 = -9" (on board), what would I do to both sides to get the x by itself (have class answer). I would just subtract 3 from both sides: x + 3 = -9 (on board) -3 -3 ----------- x = -12 Again, I immediately know what x is. B. Multiplication Principle The addition principle allows us to "undo" addition to or subtraction from the variable, so that we can get the variable by itself. Is there a similar principle for multiplication/division? Yes! If we take the equation "2 = 2", note that we can multiply both sides by, for example, 1/3. 2 . 1/3 = 2 . 1/3 (on board) 2/3 = 2/3 Thus, multiplying both sides of a true equation by the same number generates another true equation. This is called the multiplication principle. OK, now let's try applying it to variables. Consider "4x = 32" (on board). Suppose I multiply both sides by 1/4: 1/4 . 4x = 1/4 . 32 (on board) x = 8 Just as we did with addition, we've cancelled out the 4 that was multiplying the x and gotten x by itself. To get rid of a multipication by four, I divide by four, or, equivalently, multiply by one fourth. Suppose I had "-6x = 15" (on board) Here my x is multiplied by -6. What should I do to get x by itself? (let class answer) Right: I multiply by -1/6, or divide by -6. Notice that I need to multiply or divide the -6 by another negative to leave behind a positive x. (-1/6) . -6x = (-1/6) . 15 x = -15/6 = -5/2 So we've got x again. C. Addition and Multiplication Together The final step is to combine these two principles. Consider the equation "3x + 8 = 17" (on board). We want to figure out x, so we need to get x by itself. In this equation, x is both muliplied by 3 and added to 8. We need to undo both, but which first? We can be guided by the order of operations. If we look at "3x + 8", the order of operations tells us that we would do the multiply first, then the addition. To undo these, we need to work in reverse order, so we undo the addition first, and then the multiplication. So how do we undo the addition? (let class answer) We subtract 8 from both sides: 3x + 8 = 17 - 8 -8 ----------- 3x = 9 OK, now we've got just multiplication. So let's undo that. What should we do to both sides? (let class answer) We divide both sides by 3, or multiply them by 1/3: 1/3 . 3x = 1/3 . 9 x = 3 And thus we have the answer. We can check to make sure this is right by plugging in to the original equation: "3x + 8 = 17". If we plug in a 3 for x, we get: 3x + 8 = 17 3(3) + 8 = 17 9 + 8 = 17 17 = 17 So this checks out, which means that 3 really is the right value for x. II. Commutative, Associative, and Distributive Properties A. Commutative property One of the most basic ways to rewrite algebraic expressions is to change the order of certain operations. Here's what that means. Supose the we have the rule that the cost of an item is the sticker price plus the tax: cost = s + t Thus if the sticker price were $100 and the tax were $5, we would have cost = 100 + 5 = 105. Clearly, however, we would have gotten exactly the same result if we put the numbers in the opposite order: cost = 5 + 100 = 105. This is true for any sticker price and any tax. Thus we can write the rule in two equivalent ways: cost = s + t = t + s The order in which we add the two numbers doesn't matter. This goes by the fancy name of the commutative property of arithmetic. It just means that it doesn't matter what order you add in. Multiplication is the same way. Suppose we have a rule that the tax is equal to the tax rate times the sticker price. Thus tax = r . s. Continuing our example, if the sticker price is $100 and the tax rate is 5% (or 0.05), we have tax = 0.05 . 100 = $5. However, we could equally well have multiplied in the opposite order: tax = 100 . 0.05 = $5. Thus we have the rule tax = r . s = s . r. Multiplication is commutative just like addition. To put this formally, any algebraic expression that involves the addition of two numbers or letters, or the multiplication of two numbers or letters, can be replaced by an equivalent one in which the order is switched: a + b = b + a for any a and b a . b = b . a for any a and b What about subtraction? Clearly 5 - 3 is not the same as 3 - 5, so subtraction is not commutative. However, we can think of it as commutative if we think of subtraction as adding a negative: 5 - 3 = 5 + (-3) = (-3) + 5 = 2 Thus we can also say that a - b = a + (-b) = -b + a for any a and b. B. Associative property Another closely-related rule we can use to find equivalent algebraic expressions involves the placement of parentheses. Since the order doesn't matter for addition or multiplication, and parentheses are just rules about order, we can move them around freely. Here's an example: suppose the rule we have is that a burrito shop's monthly income is the number of burritos it sells times the price per burrito: income = b p. One month it has a special where it sells the burritos for 20% off, and as a result it sells twice as many as usual. We can use our formula to figure out how much money the store makes compared to in a usual month: income = (2 b) (0.8 p) The b becomes 2b because the store sold twice as many burritos, but each one only made 80% as much as usual. These are all multliplicative operations, and since the order doesn't matter, we can move around the parentheses: income = (2b) (0.8p) = (2 . 0.8) (bp) = 1.6 bp. Thus we come up with a simpler expression, and see that the shop makes 1.6 times as much as it would in a normal month. The general rule here is that a(bc) = (ab)c for any a, b, and c Exactly the same argument applies to addition: a + (b + c) = (a + b) + c Here's one for practice: simplify (2x)(3y)(4x) C. Distributive property We started talking about this last week. To remind you: distributing terms. Distribution means breaking up an algebraic expression into simpler pieces. For example, suppose we had the expression 4 (x + y) The 4 gets multiplied by x and y, so we can simplify by writing out the multplication explicitly: 4 (x + y) = 4x + 4y This is called distributing the 4 over the x and y. Distribution works over any number of items (called terms) inside the parentheses: 4 (x + y + z) = 4x + 4y + 4z As with the commutative property, we can apply this with signs too, as long as we're careful. A minus sign is equivalent to multiplying by -1. Thus -(x + 1) = (-1) (x + y) = (-1)x + (-1)y = -x - y. Thus if a minus sign appears in front of a parentheses, it must be applied to every term inside the parentheses. Here's one for practice: what is -2 (a - 2b)? We can also distribute division using a similar trick: recall that dividing by a number is the same as multiplying by 1 over that number. For example, 8 / 2 is the same as (1/2) . 8. Thus we can write (x + y) / a = x/a + y/a for any x, y, and a Here's a practice example: what is (-x+y)/(-2)? III. Factoring out common factors The commutative, associative, and distributive properties are the basic rules for manipulating algebraic expressions. We can use them to construct more complex and powerful rules. Many of these fall under the heading of factoring. Factoring is a technique for taking an algebraic expression and rewriting it as an equivalent multiplication of simpler expressions. Next week we'll go deeper into factoring, but for now we'll start with its most basic application: taking out common factors. We have seen that the distributive property is a(b + c) = ab + ac. The simplest form of factoring is just to do this in reverse when we notice that two terms of a factor in common. Here's a simple example: in the expression 2x + 2y, both terms have a factor of 2. Thus we can rewrite this: 2x + 2y = 2(x + y). Here's a somewhat trickier one: 2x + xy = x (2 + y). The idea is the same, and it's just that this time the common factor was a letter instead of a number. As another example, consider (on board) 2x^4 + 4x^3 + 6x^2 We can notice that all of the monomials in this polynomial contain an x. We call x a common factor. Thus we can re-write this as (on board) 2x^4 + 4x^3 + 6x^2 = x (2x^3 + 4x^2 + 6x) Just by looking at this expression, it is appearant that if I were to multiply out the right-hand side I would get back to the original expression. The idea here is to find a factor that all the terms have in common, and pull it out front. In the expression we have now, we could in fact factor it further. Can anyone suggest something else that these terms have in common? (let class answer) (on board) 2x^4 + 4x^3 + 6x^2 = x (2x^3 + 4x^2 + 6x) = (x)(x) (2x^2 + 4x + 6) = x^2 (2x^2 + 4x + 6) We can pull out another factor of x. If we'd seen this originally, we could just have pulled out a factor of x^2. In fact, there's one more common factor we can pull out. Can anyone see it? (let class answer) (on board) 2x^4 + 4x^3 + 6x^2 = x (2x^3 + 4x^2 + 6x) = (x)(x) (2x^2 + 4x + 6) = x^2 (2x^2 + 4x + 6) = (x^2) (2) (x^2 + 2x + 3) = 2x^2 (x^2 + 2x + 3) Once we've pull out the full 2x^2, there's no way to pull out more common terms. There is no number or variable expression that goes into each monomial term we have left.