Lesson 6: Algebra Chapter 2, Video 1: "Variables" Algebra 1, variables. In math, when the value of a number isn't known, a letter is used to represent the unknown number. This letter is called a variable. A variable can be any letter in either uppercase or lowercase form: x, y, z, a, n, capital X, capital Y, capital R, et cetera. But the lowercase x is used most of the time. Multiplying with variables. How might you express an unknown number that is multiplied by itself four times? You just learned that if a number is unknown, you express it as a variable. Again, this variable can be any letter you want, but let's just use x for now. So x is multiplied by itself four times, which means you have the following: x times x times x times x equals x to the fourth. This is done in the same way that if you were asked to express 3 being multiplied by itself 4 times. Do you see how the process is the same? The only difference is that this can be simplified to 81. However, with the variable, we would have to leave it as x to the fourth power because we don't know what x is. So, what would y times y times y equal? That's right, y to the third. When multiplying variables that are different, you'll just combine the variables. For example, a times b equals ab. What about m times n times p? That would give you this. Do you see the pattern? Again, all you need to do is combine the variables. What about when you have numbers and variables? In a similar manner, all you do is combine the number and variable, and this would be your answer. Let's try another problem. 8x times 3y. In this case, multiply the numbers the way you normally would, and then multiply the variables by combining them like you just learned. So this would be your answer: 8x times 3y equals 24xy. Let's try one more. 9q times r times -2q equals what? Again, you would begin by multiplying the numbers. However, I'm sure that you noticed r has no number with it. When there's no number in front of a variable, you can think of the variable as having an invisible 1 in front of it. After all, 1 times r equals r. Multiply the numbers; 9 times 1 times -2 equals -18. Multiply the variables, q times r times q. q times q equals q squared. q squared times r equals q squared r. Your final answer is 9q times r times -2q equals -18q squared r. Combining like terms. When adding and subtracting variables, you must have exactly the same variables with the same power present in order to combine the terms. If exactly the same variables with the same powers are present, the terms are considered to be alike, hence the phrase, "combining like terms." Here's an example. 3x plus 4x equals what? The 3 has just an x with it and the 4 has just an x, so the 3x and 4x are considered like terms, which is why you are able to add them together. And just like 3 plus 4 equals 7, 3x plus 4x equals 7x. Let's try another. 8a minus 10a equals what? Again, you have like terms, because the 8 has just an a with it, as does the 10. And just like 8 minus 10 equals -2, 8a minus 10a equals -2a. Let's try another. 4x plus 2y minus 8 minus 4x plus 9y minus 1 equals what? This one's a little more complicated. Let's take it step by step. You know that you can only combine x's with x's and y's with y's. Make sure that you pay attention to the sign in front of your term. If there's a
plus sign in front of the term, add that term, and if there's a minus sign in front, subtract that term. Since there's nothing left to combine, we're done. Let's try one more. 3a squared plus 2a squared b squared minus 5a squared plus 3ab squared. The only terms that have the exact same variable and exponent are 3a squared and -5a squared. 3a squared minus 5a squared equals -2a squared. So 3a squared plus 2a squared b squared minus 5a squared plus 3ab squared equals -2a squared plus 2a squared b squared, plus 3ab squared. Are you wondering what happens if there are no like terms to combine? Let's find out by taking a look at the following: 8v plus 8 equals what? In this case, there are no v's to combine and no numbers to combine either, so you would just leave this as is, because there's nothing you can compute. Would you be able to combine any terms in the following expression: xy plus x minus x squared y minus y squared plus xyz? No, you wouldn't. None of the terms are alike. Remember, letters, their combinations, and their powers must be the same in order for them to be like terms and therefore combinable. Note, make sure that you don't confuse adding variables and multiplying variables. x plus x equals 2x. x times x equals x squared. Or 8 plus y equals 8 plus y. 8 times y equals 8y. Math properties. The first math property you're going to learn about is called the commutative property. The commutative property applies to addition and multiplication. It basically states that the order in which you add or multiply has no effect on the answer. So for addition, a plus b equals b plus a. Let's take a closer look with an example. 2 plus 7 equals 7 plus 2, right? Both will give you 9, regardless of the order. The same applies for multiplication. ab equals ba. Here's our example. 3 times 4 equals 4 times 3, equals 12. Again, regardless of the order, you will still end up with the same answer. The next property is called the associative property, and it also applies to both addition and multiplication. The associative property states that when adding or multiplying three or more numbers, the answer will remain the same, regardless of how they are grouped together. Let's take a look. Associative property of addition. Example. Associative property of multiplication. Example. Do you see how in both cases, the grouping doesn't matter? Either way, you will get the same answer. Our next property is the distributive property. This property involves distributing one term to two or more other terms. a times b plus c equals ab plus ac. Here's an example. 3 times r plus t equals 3 times r plus 3 times t, equals 3r plus 3t. You take the term on the outside of the parentheses in this case, the number 3 and distribute it by multiplying to the terms inside of the parentheses, the r and the t. Let's try another example. 6 times x plus y minus 1 equals 6 times x plus 6 times y minus 6 times 1, equals 6x plus 6y minus 6. Since you have no like terms to combine, 6x plus 6y minus 6 is your final answer. There may be times when you have a negative number to distribute. When this happens, you must take that into account so that you get the correct answer. Let's try a problem. -5 times 3c
plus 1 minus d equals -5 times 3c plus -5 times 1 minus -5 times d, equals -15c minus 5 plus 5d. There are no like terms, so your final answer is -15c minus 5 plus 5d. The last property you need to know is the substitution property. The substitution property means that if you have two values that are equal, you can substitute one value for the other. Let's say you want to calculate 3x and you were told that x equals -4. To calculate the value of 3x, you can substitute the -4 in for x and then find the answer. 3x, or 3 times -4, equals -12. As you can see with substitution, you just plug in for the variables you were given and simplify, if possible. Give this problem a try. Find the value of 7y, when y equals 5z. 7y equals 7 times 5z, equals 35z. By plugging in 5z for y, you end up with 7 times 5z, which gives you an answer of 35z.
Lesson 6: Algebra Chapter 3, Video 1: "Algebra 2" Algebra 2. One of the most important aspects of algebra involves solving equations. An equation is a mathematical statement in which two expressions are shown as being equal to one another. Many of the equations you'll work with will contain variables, which, in most cases, you will need to solve for. The following is an example of an equation: 8x equals 40. Notice how there is an expression on each side of the equals sign, and the variable x needs to be solved. The correct value of x will make both sides of the equation equal to each other. So, in this case, when x equals 5, the equation holds true, because you'd have the following: 8x equals 40. x equals 5. 8 times 5 equals 40. 40 equals 40. Any value of x besides 5 would cause the two sides not to be equal. For example, if you had x equals 10, you would end up with 80 equals 40, which is not true. So when you solve equations, you are finding the value or values of the variables that make the two sides equal to each other. Solving equations. Let's try solving some basic equations. x plus 7 equals 15. I'm sure you can look at this and know that x is 8, but let's go through it so you can build a good foundation in solving equations. When solving equations, the goal is to get the variable alone on one side of the equation. That way you'll know what its value is, because your final answer will be in the form of x equals some number. The way to get the variable alone is to manipulate the equation by performing the opposite operations of the ones that are present in the equation. So this is what we're starting with: x plus 7 equals 15. At the moment, the 7 is connected to the x by addition, which we know because there is a plus-7 attached to the x. To get the x alone, you have to get rid of the 7. How? You perform the opposite operation. If you subtract 7 from 7, you'll end up with 0, which is what you want, because x plus 0 equals x, and that gives you the x alone. However, you can't just subtract 7 from one side of the equation. What you do to one side, you must always do to the other side. This is because the two sides of an equation are equal to one another. If you only change one side, you are not keeping the two sides equal. Let's go ahead and solve the equation. Subtract 7 from both sides. And you can always check your answer by substituting the value of your variable into your original equation to make sure that your answer is correct. 15 equals 15 is true, so the answer is correct. Let's solve another equation. -10 equals x minus 8. In this example, the variable is on the right side of the equals sign. This doesn't matter though. The equation still states that the two sides are equal to each other and the goal and process are the same. You still want to get x alone, and you still need to perform the opposite operation to do so. This is how we solve the equation. Add 8 to both sides. Then you can check your answer by plugging in -2 for x in the original equation to see if it makes the equation true. -10 equals -10, so the answer is correct. Let's try some algebra problems that involve multiplication and division. 6y equals 42. Your immediate goal is still to get y alone, and you still need to perform the opposite operation to do so. 6y means that the 6 is connected to the y by multiplication. The opposite operation would
be division in this case. Divide both sides by 6 to undo the multiplication. The 6's cancel out, since 6 divided by 6 equals 1. Let's check our answer. Let's try another example. x divided by 3 equals 8. You're dividing the x by 3. That means to get the x by itself, you'll need to multiply by 3. Multiplication is the opposite of division. Let's finish the problem. Multiply both sides by 3. The 3's cancel out, because 3 divided by 3 is 1. Check your answer. Let's try solving a slightly more difficult equation. 3x plus 7 equals 16. In this case, we have to get rid of the 7 that's connected by addition and the 3 that's connected by multiplication. First, we'll get rid of what's furthest from x, and that's the 7. To undo addition, we subtract, and what we do to one side, we do to the other. So we have this. To undo the multiplication of the 3 to the x, we divide by 3. Check. So our answer of 3 is correct. Complex equations. Sometimes you'll have more than one operation to perform in an equation. You also may have to distribute or bring variables over to the other side of an equation. Just know that the goal and the process are always the same. You always want to get the variable that you're solving for alone on one side of the equation. It doesn't matter which side. And you always do the opposite operation to undo an operation that is already there. Let's try a few examples. 4 times x minus 1 plus 8 equals 36. Your first step is to distribute. You don't want anything in parentheses. Then combine like terms on each side. In this case, we can only combine the -4 and the 8. Next, subtract 4 from both sides. We're almost done getting the x alone. Our next step is to divide both sides by 4. Now we'll cancel out the 4's. Divide 32 by 4, and you have your answer. x equals 8. The final step is to check your answer. Now that we've had a little practice, let's go over the steps to solving the algebraic equations you'll see on the test. Some of these steps may not be necessary, depending on the equation that you're solving. If a step is not necessary, just move on to the next step until you've solved for the variable. 1.) Distribute. 2.) Combine like terms on each side of the equation. 3.) Add or subtract numbers and variables so that you have variables on one side of the equals sign, and numbers on the other side. 4.) Multiply or divide to get x alone. Let's try one more problem to see if this is making sense. 5x plus 5 minus x equals x minus 10. Combine like terms on each side of the equation. Subtract variables to get them on one side of the equation. Subtract 5 from both sides to begin getting your numbers on the other side of the equation. Divide both sides by 3 to get x alone. Check your answer. Word problem equations. You may also come across problems where there's an equation to be solved, but instead of the equation being given in algebraic form, it's in word form. It will be up to you to convert the equation to algebraic form and solve from there. Let's try a few examples. The product of two numbers is 80. If one of the numbers is 8, what's the other number? Right
away, we know that we have two numbers that when multiplied together equal 80, because product means the result of multiplication. We know that one of the numbers is 8. Since we don't know what the other number is, we'll call it x. Let's set up our equation. 8x equals 80. Now let's solve. So our missing number is 10. Since 8 times 10 equals 80, this is correct. Let's try another problem. The sum of three consecutive integers is equal to 45. What are the three integers? We know that integers include positive and negative whole numbers and zero. What does consecutive mean? It means that these numbers will be in order, one after the other. For example, 1, 2, 3, 4 are consecutive integers. How do we set up our equation? We know that sum means addition, so we're adding these three numbers. Since we don't know the first number, that will be our x. And after x would come x plus 1. Our third number would be x plus 2. Here's how you set up this equation. So our first number's 14, which means our second and third numbers are 15 and 16. We can check our answer quickly by adding the numbers together. Literal equations. Literal equations are equations with several variables. The steps are the same when solving them. However, since the variables are different, you're not going to be able to combine like you normally would with numbers. Let's try an example. F equals ma, and you're solving for a. Since m is connected to a by multiplication, to get a alone, you need to divide by m. Even though you're not able to combine F and m since they're different variables, you have now solved for a, since a is alone. Let's try one more. ax plus b equals c. We're solving for x. First, subtract b from both sides. Then divide both sides by a to get x alone. Inequalities. An inequality is a mathematical statement in which two expressions are shown to be unequal to one another. The way to tell the difference between the less-than and greaterthan part of the symbol is to determine whether there is an open end or a closed end. The open end is the greater-than part, and the closed end is the less-than part. For example, 2 is less than 8, or 8 is greater than 2. When solving an inequality, the rules are pretty much the same as when you have an equation. Earlier, we solved the equation 5x plus 5 minus x equals x minus 10. If this were an inequality, we would solve it exactly the same way we solved it as an equation. Let's take a look. Combine like terms on each side of the equation. Subtract variables to get them on one side of the equation. Subtract 5 from both sides to begin getting your numbers on the other side of the equation. Divide both sides by 3 to get x alone. So when we solve an equation, we are finding a value for the variable that makes both sides equal, making the equation true. When we solve an inequality, we are finding the values for the variable that make the inequality statement true. In this case, all values greater than or equal to -5 will make this inequality true. How do we check this? Just choose a number greater than or equal to -5 and see if the inequality statement holds. Let's choose 0 and see what happens. Is 5 greater than -10? It is, so we solved this inequality correctly. There is one instance when solving an inequality won't be the exact same as solving an equation. This would be when you have a negative number attached to the variable you're
solving for. In this case, your last step would be where you have to multiply or divide by a negative number to get the variable alone, and because of this, you must change the direction of the inequality sign. Let's look at an example. Your first step is to subtract 5 from both sides. Then, since you have to divide by a -2 to get x alone, you must flip the direction of the inequality sign. So it becomes... Check this. Choose a number less than -2 and substitute that in for x. Let's use -3. This is true. 11 is greater than 9, so our solution is correct.