WEBVTT 00:21.680 --> 00:28.596 In this video, we're going to study Properties of Equality. The first Property of Equality that we're going to consider 00:28.629 --> 00:41.142 is the Addition Property of Equality. It says that if a, b, and c are real numbers and a is equal to b, then a plus c is equal 00:41.200 --> 00:49.784 to b plus c. This means that I can add the same number to both sides of this equal sign 00:49.840 --> 00:56.657 and still have a true statement. Because subtraction can also be written as an 00:56.720 --> 01:04.031 addition statement, plus a negative number, this property holds true for subtraction as well. 01:04.080 --> 01:14.909 And I can write that if a equals b then a minus c is equal to b minus c. 01:14.960 --> 01:20.581 Let's look at these examples. The directions are to solve the equation. 01:20.640 --> 01:28.989 To solve an equation, we're going to isolate the variable on one side of the equal sign 01:29.040 --> 01:34.495 Using the Addition Property of Equality, I can rewrite this equation to an equivalent 01:34.560 --> 01:41.302 equation that has the same set of solutions as this original equation. Notice that on the 01:41.360 --> 01:49.877 right side we have x plus one-half. To isolate the x, I need to get rid of the plus one-half. 01:49.920 --> 02:04.124 So we will subtract one-half from both sides of the equal sign. On the right-hand side, I can see that plus one-half 02:04.160 --> 02:13.167 minus one-half would be zero. So I'm left with x plus zero, or simply x. 02:13.200 --> 02:24.678 And on the left-hand side, we have three-fourths minus one-half. Now remember, to add or subtract 02:24.720 --> 02:31.040 fractions, you must have a common denominator. The lowest common denominator of four and two 02:31.051 --> 02:38.926 is four. So I need to multiply this fraction by two over two. That does not change the value 02:38.960 --> 02:47.134 because I'm simply multiplying by one. This now gives me two over four, so I have three 02:47.200 --> 03:00.681 over four minus two over four, leaving us with one over four. So my solution is x equals one-fourth. 03:00.720 --> 03:12.459 This solution is the solution to the original equation. The next example we have is x minus one point two is equal to 03:12.493 --> 03:22.336 three point five. Again, to solve this equation, I want x isolated on one side of the equal sign. So this minus 03:22.400 --> 03:35.215 one point two needs to move. I'm going to add one point two to both sides of this equation. 03:35.280 --> 03:39.720 By the Addition Property of Equality, when I add the same thing to both sides of the equation, 03:39.760 --> 03:50.030 I don't change the solutions. And so negative one point two plus one point two is zero. 03:50.080 --> 03:57.304 x plus or minus zero is just x. And on the right-hand side, three point five plus one 03:57.360 --> 04:10.818 point two is four point seven. So our solution to this equation is x equals four point seven. 04:10.880 --> 04:17.791 Our next example is seven a minus five equals eight a. Again, we are trying to solve 04:17.840 --> 04:23.564 the equation which means we want to isolate the variable on one side of the equal sign. 04:23.600 --> 04:30.504 It does not matter whether I isolate the variable on the left or on the right. In this case, 04:30.560 --> 04:35.943 I'm going to choose to isolate the variable on the right side as a slightly more efficient method of 04:36.000 --> 04:43.383 solving this equation to reduce the number of operations that I have. So if I want to 04:43.440 --> 04:59.066 move the seven a to the right side, I am going to subtract seven a from both sides of this equation. 04:59.120 --> 05:06.206 Notice that seven a minus seven a is zero and zero minus five is negative five. So on the left, 05:06.240 --> 05:20.020 we have negative five. On the right, we have eight a minus seven a which is one a, or just a. 05:20.080 --> 05:27.995 This is an equivalent equation to the equation that was given. It's in a more convenient form because 05:28.028 --> 05:42.109 now we know exactly the value of a. a is negative five. The solution to the original equation is negative five. 05:42.160 --> 05:53.320 In our next example, we have one-fourth m plus seven-fourths m equals m plus five-sixths. 05:53.360 --> 06:02.896 We are solving this equation. So we want to have all of the m terms on one side of the equation. 06:02.960 --> 06:08.335 First however, I'm going to notice that I can combine these terms here using 06:08.400 --> 06:18.512 the Distributive Property. This sum can be rewritten as one-fourth plus seven-fourths 06:18.560 --> 06:31.625 times m. And that is equal to m plus five-sixths. One-fourth plus seven-fourths is eight-fourths, 06:31.680 --> 06:47.007 and eight-fourths is equivalent to two. So this left side is two m. The right side is m plus five-sixths. 06:47.040 --> 06:59.086 Now, I'm going to move the m from the right side to the left by subtracting m from both sides. 06:59.120 --> 07:09.296 On the right side, m minus m is zero, and zero plus five-sixths is five-sixths. 07:09.360 --> 07:28.415 And on the left side, two m minus m is m. So we have m equals five-sixths. This is the solution to the original equation. 07:28.480 --> 07:35.322 The Multiplication Property of Equality states that if a, b, and c are real numbers, 07:35.360 --> 07:43.797 c is not equal to zero, and a is equal to b, then a times c is equal to b times c. 07:43.840 --> 07:49.469 Because division can also be rewritten as a multiplication problem, this property holds true 07:49.520 --> 08:04.918 for division as well as long as c is not equal to zero. If a is equal to b, then a divided by c is equal to b divided by c. 08:04.960 --> 08:14.294 We'll now use this property to solve the following equations. The first equation is two-thirds x equals five. 08:14.327 --> 08:25.205 We want to isolate x on one side of the equal sign. Currently, it is being multiplied times two-thirds. I can divide 08:25.238 --> 08:36.516 both sides of this equation by two-thirds, or I can also multiply both sides of this equation by the reciprocal of two-thirds. 08:36.560 --> 08:43.457 The reciprocal of two-thirds is three-halves. So we will multiply both sides of this equation 08:43.520 --> 08:55.135 by three-halves. When you multiply a number times its reciprocal, the product is one. Notice 08:55.200 --> 09:05.312 six divided by six would be one. So the left side of this equation is one x, or just x. 09:05.360 --> 09:15.789 The right side of this equation is five times three-halves which is fifteen-halves. 09:15.840 --> 09:27.634 So the solution to our original equation, two-thirds x equals five, is fifteen-halves. In our next equation, we have point four 09:27.680 --> 09:35.242 x equals point eight eight. The point four is being multiplied times x, 09:35.280 --> 09:46.353 so we will divide both sides by point four. So that we will get x remaining on the left side. 09:46.386 --> 09:56.263 Dividing both sides of this equation by point four. On the left side, point four divided by point four is one, 09:56.296 --> 10:10.277 and one x is simply x. And on the right-side, point eight eight divided by point four is two point two. 10:10.320 --> 10:19.085 So the solution to our equation, point four x equals point eight eight, is two point two. 10:19.120 --> 10:24.124 In some problems, it is necessary to use both the Addition Property of Equality and the 10:24.160 --> 10:31.398 Multiplication Property of Equality to solve the equation. Let's look at these next few examples. 10:31.440 --> 10:37.637 We're going to solve each equation. In the first example, we have seven-eighths a 10:37.680 --> 10:45.078 is equal to eight-sevenths a. We're going to begin by using the Addition Property of Equality 10:45.120 --> 11:00.026 to get both terms with a on the left side of the equation. So we're going to subtract eight-sevenths a from both sides. 11:00.060 --> 11:12.439 On the right side of the equation, we have eight-sevenths a minus eight-sevenths a which is zero. 11:12.480 --> 11:19.245 On the left side of the equation, we have seven-eighths a minus eight-sevenths a. 11:19.280 --> 11:24.584 I’m going to rewrite this subtraction using the Distributive Property. 11:24.640 --> 11:35.562 So this becomes seven-eighths minus eight-sevenths a is equal to zero. 11:35.600 --> 11:43.470 Now, I could subtract these fractions and come up with an equivalent answer for that difference, 11:43.503 --> 11:53.980 and then divide both sides by that number. But first let's think about this problem. I'm going to want to isolate a on the left, 11:54.013 --> 12:03.156 so I need to divide by this number, whatever this number is. So if I divide both sides of this equation 12:03.200 --> 12:17.871 by this number, I’m going to have zero divided by some number. Zero divided by a non-zero number 12:17.920 --> 12:28.014 is just zero. So this divided on both sides is still going to give me zero. The solution 12:28.080 --> 12:39.058 to the equation seven-eighths a equals eight-sevenths a is zero. 12:39.120 --> 12:48.101 In the next example, we have six-fifths y minus five is equal to negative seven. We want to solve 12:48.160 --> 12:55.775 the equation, meaning that we want to isolate the variable y on one side of the equal sign. 12:55.840 --> 13:05.718 So we can see that we have a six-fifths multiplied times y, and we also have a minus five 13:05.760 --> 13:13.459 added to this term or positive five subtracted from this term. We're going to begin by moving 13:13.520 --> 13:23.536 the minus five. We do that by adding five to both sides of the equation. 13:23.600 --> 13:34.280 On the left-hand side, I can see that minus five plus five is zero, leaving us with six-fifths y. 13:34.320 --> 13:40.520 And on the right-hand side of the equation, we have minus seven plus five which is a negative 13:40.560 --> 13:49.629 two. Now I have six-fifths multiplied times y. We'll use the Multiplication Property of Equality 13:49.680 --> 13:56.603 and multiply both sides of this equation by the reciprocal of six-fifths. The reciprocal of 13:56.640 --> 14:09.616 six-fifths is five-sixths. So we will multiply both sides of the equation by five-sixths. 14:09.680 --> 14:18.358 Five-sixths times six-fifths is one, leaving us with one y, or just y, on the left. 14:18.400 --> 14:35.074 Negative two times five-sixths gives us negative five-thirds on the right. The solution to this equation is negative five-thirds. 14:35.120 --> 14:42.982 In our last example, we have fifteen x minus nine is equal to five x plus nineteen. 14:43.040 --> 14:48.654 We want to solve the equation, so we want all of the variables on one side of the equation. 14:48.720 --> 14:56.429 So I'm going to begin by subtracting five x from both sides of the equation, 14:56.480 --> 15:05.004 essentially moving this term to the left side. I notice that I’m also going to want to move this 15:05.040 --> 15:09.709 minus nine because it does not contain an x. I want to move that to the right side 15:09.760 --> 15:21.220 of the equation, so I will also add nine to both sides of this equation. 15:21.280 --> 15:29.462 Here I have minus fifteen x minus five x, excuse me I have fifteen x minus five x. 15:29.520 --> 15:36.202 I'm going to use the Distributive Property to write this as fifteen minus five 15:36.240 --> 15:48.414 times x. We have minus nine plus nine which is zero equals five x minus five x is zero, 15:48.480 --> 15:59.859 and nineteen plus nine is twenty-eight. Fifteen minus five is ten, so on the left side I have ten 15:59.920 --> 16:08.434 x. And on the right side, I have twenty-eight. Now using the Multiplication Property of Equality 16:08.480 --> 16:20.279 which also holds for division, I'm going to divide both sides of this equation by ten. 16:20.320 --> 16:27.954 Ten divided by ten is one, leaving me with one x, or just x. And twenty-eight 16:28.000 --> 16:35.394 divided by ten can be simplified by dividing both the numerator and denominator by two, 16:35.440 --> 16:50.510 leaving us with fourteen-fifths. The solution to this original equation is fourteen-fifths. 16:50.560 --> 16:58.584 All of the equations that we have looked at and solved in this section have been linear equations 16:58.640 --> 17:06.025 in one variable. Let's look at the definition of a linear equation in one variable. A linear 17:06.080 --> 17:15.401 equation in one variable is an equation that is written in the form ax plus b equals c where a, b, 17:15.440 --> 17:26.345 and c are real numbers and a is not equal to zero. Notice a cannot be equal to zero here 17:26.400 --> 17:34.921 because then zero times x would be zero, and you would no longer have a variable in the equation, 17:34.960 --> 17:40.993 or a linear equation in one variable can be an equation that can be written 17:41.040 --> 17:48.901 in this form using properties of equality or properties of arithmetic. Properties of 17:48.960 --> 17:55.641 equality include the Multiplication Property of Equality and the Addition Property of Equality. 17:55.680 --> 18:00.179 The properties of arithmetic that we have studied include the Commutative Property, 18:00.240 --> 18:13.125 the Associative Property, and the Distributive Property. The form of this equation a x plus b equals c is a linear equation 18:13.159 --> 18:23.135 in one variable. Notice that the variable x here has an exponent of one. We don't see the exponent of one. 18:23.169 --> 18:31.544 It's implied. So any equation that has x to any other power besides one is not a linear equation in one 18:31.600 --> 18:40.353 variable. If the x is underneath of a radical sign or in the denominator of a fraction, that equation 18:40.400 --> 18:55.840 also is not a linear equation. ax plus b equals c is the form of a linear equation in one variable.