WEBVTT 00:00:02.096 --> 00:00:07.806 A:middle >> Welcome to the Cypress College math review on solving simple linear equations. 00:00:08.956 --> 00:00:12.416 A:middle Objective one. 00:00:12.786 --> 00:00:17.116 A:middle Solving linear equations using properties of equality. 00:00:17.606 --> 00:00:19.536 A:middle So we're trying to solve an equation. 00:00:20.336 --> 00:00:26.596 A:middle So our goal is to get x on one side, isolate it, and simplify the other side. 00:00:27.576 --> 00:00:29.156 A:middle We're going to use the addition 00:00:29.156 --> 00:00:32.386 A:middle and multiplication properties of equality to help us do this. 00:00:33.046 --> 00:00:35.456 A:middle Let's say that a, b and c are real numbers. 00:00:36.926 --> 00:00:41.466 A:middle This property a plus c equals b plus c is equivalent 00:00:41.466 --> 00:00:45.046 A:middle to a equals b is called the addition property of equality. 00:00:46.236 --> 00:00:50.276 A:middle This property tells us that you're allowed to add 00:00:50.276 --> 00:00:54.126 A:middle or subtract the same quantity to both sides of an equation. 00:00:56.156 --> 00:01:03.086 A:middle ac equals bc is equivalent to ab is the multiplication property of equality. 00:01:03.856 --> 00:01:09.436 A:middle This tells us that you can multiply or divide the same quantity to both sides of an equation. 00:01:09.666 --> 00:01:13.376 A:middle We will use these two properties in solving equations. 00:01:19.316 --> 00:01:23.516 A:middle Example. Solve for x. x minus 7 equals 3. 00:01:24.246 --> 00:01:29.756 A:middle So to solve for x, we need to get x by itself, so we need to get rid of this minus 7 deal. 00:01:30.086 --> 00:01:31.406 A:middle What do we do? 00:01:31.746 --> 00:01:35.516 A:middle Well, we use the addition property of equality and we add 7. 00:01:35.786 --> 00:01:39.736 A:middle Well, if you do it to one side, you must do the exact same thing to the other side. 00:01:40.076 --> 00:01:42.846 A:middle So we need to add 7 to both sides of the equation. 00:01:43.476 --> 00:01:51.296 A:middle Notice that minus 7 and plus 7 add up to 0 so that disappears and 3 plus 7 is equal to 10. 00:01:51.856 --> 00:01:57.006 A:middle So on the left-hand side, all we have left is x. And on the right-hand side we have 10. 00:01:57.796 --> 00:01:59.976 A:middle Therefore, x equals 10 and it's solved. 00:02:05.236 --> 00:02:09.556 A:middle Example. Solve for x. 5 plus x equals 1. 00:02:09.806 --> 00:02:13.176 A:middle Well, we wish to get x by itself, so we need to get rid of the 5. 00:02:13.906 --> 00:02:16.996 A:middle That's a positive 5 out there and that's a separate term 00:02:16.996 --> 00:02:20.086 A:middle from the x so we would need to subtract 5. 00:02:20.466 --> 00:02:23.076 A:middle If we do that to the left side, we must do it to the right side. 00:02:23.406 --> 00:02:25.476 A:middle So we subtract 5 from both sides. 00:02:25.936 --> 00:02:28.776 A:middle A positive 5 and a negative 5 add up to 0. 00:02:29.186 --> 00:02:35.506 A:middle So the only thing left on the left-hand side is x. 1 minus 5 is negative 4. 00:02:35.996 --> 00:02:39.816 A:middle So x is equal to negative 4, and we have our solution. 00:02:44.636 --> 00:02:49.206 A:middle Example. Solve for x. x divided by 8 equals 3. 00:02:49.526 --> 00:02:53.056 A:middle So we wish to isolate the x. So we need to get rid of 8. 00:02:54.186 --> 00:02:56.126 A:middle x is being divided by 8. 00:02:56.386 --> 00:03:00.156 A:middle So the inverse operation to dividing is multiply. 00:03:00.636 --> 00:03:05.936 A:middle So we need to use the multiplication property of equality and multiply both sides by 8. 00:03:06.556 --> 00:03:12.146 A:middle A lot of students, when they're multiplying fractions, like to write the 8 as 8 over 1. 00:03:12.726 --> 00:03:16.446 A:middle So now you see that the 8s cancel. 00:03:16.636 --> 00:03:19.006 A:middle So we get x on the left-hand side. 00:03:19.006 --> 00:03:26.196 A:middle x over 1 is just x. And then on the right-hand side we have 8 times 3 and that gives us 24. 00:03:26.506 --> 00:03:27.976 A:middle So x is equal to 24. 00:03:33.126 --> 00:03:36.186 A:middle 5 times x equals negative 20. 00:03:37.196 --> 00:03:40.346 A:middle So 5 is being multiplied times the x this time. 00:03:40.586 --> 00:03:43.616 A:middle The inverse operation to multiply is divide. 00:03:44.516 --> 00:03:46.426 A:middle So we divide both sides by 5. 00:03:47.046 --> 00:03:52.446 A:middle Notice that the 5s cancel on the left-hand side so we just get x. x is isolated. 00:03:52.446 --> 00:03:58.166 A:middle On the right-hand side, we have negative 20 divided by 5 and that's negative 4. 00:03:58.926 --> 00:04:01.356 A:middle Therefore, x is equal to negative 4. 00:04:01.986 --> 00:04:06.976 A:middle Pause the video and try these problems. 00:04:17.626 --> 00:04:18.856 A:middle Objective two. 00:04:19.426 --> 00:04:21.866 A:middle Solving multi-step linear equations. 00:04:22.426 --> 00:04:26.936 A:middle Example. Solve for y. The variable is not always going to be x. 00:04:27.636 --> 00:04:30.556 A:middle So our goal is to get y isolated by itself. 00:04:31.436 --> 00:04:36.086 A:middle We have negative 10 times y plus 15 equals 45. 00:04:36.586 --> 00:04:39.876 A:middle Think of your order of operations. 00:04:40.426 --> 00:04:42.836 A:middle You would multiply before you add, correct? 00:04:43.786 --> 00:04:50.556 A:middle But to solve an equation, you reverse your order of operations to isolate the variable. 00:04:51.316 --> 00:04:56.686 A:middle So we will subtract 15 before we divide by the negative 10. 00:04:56.896 --> 00:05:00.076 A:middle So first subtract 15 from both sides. 00:05:00.576 --> 00:05:05.426 A:middle Notice that positive 15 and negative 15 add up to 0. 00:05:05.916 --> 00:05:08.706 A:middle So our left-hand side is just negative 10y. 00:05:09.296 --> 00:05:13.556 A:middle On the right-hand side, we have 45 minus 15 and that's 30. 00:05:14.276 --> 00:05:18.806 A:middle Now we have negative 10 times y equals 30. 00:05:19.366 --> 00:05:24.186 A:middle The inverse operation to multiply is divide which means we need 00:05:24.186 --> 00:05:26.956 A:middle to divide both sides by negative 10. 00:05:27.576 --> 00:05:33.146 A:middle The negative 10s on the left-hand side cancel which just gives you 1y or just y. 00:05:33.656 --> 00:05:38.376 A:middle On the right-hand side, 30 divided by negative 10 is negative 3. 00:05:39.416 --> 00:05:41.976 A:middle Therefore, y is equal to negative 3. 00:05:47.256 --> 00:05:52.736 A:middle Example. Solve for x. x divided by negative 2 plus 8 is equal to 11. 00:05:53.896 --> 00:05:58.356 A:middle We wish to isolate x. We reverse our order of operations, 00:05:58.606 --> 00:06:01.566 A:middle and we need to get rid of the plus 8 first. 00:06:02.576 --> 00:06:05.466 A:middle Since it's plus 8, we would subtract 8. 00:06:06.166 --> 00:06:08.836 A:middle So we subtract 8 from both sides. 00:06:09.426 --> 00:06:11.856 A:middle Positive 8 and negative 8 add up to 0. 00:06:12.326 --> 00:06:15.446 A:middle So we just have x divided by negative 2 on the left-hand side. 00:06:16.136 --> 00:06:18.646 A:middle And 11 minus 8 is equal to 3. 00:06:20.706 --> 00:06:24.176 A:middle Now we have x divided by negative 2. 00:06:24.176 --> 00:06:26.246 A:middle We wish to get rid of the negative 2. 00:06:26.896 --> 00:06:32.646 A:middle Since x is being divided by negative 2, the inverse operation to divide is multiply. 00:06:33.036 --> 00:06:35.876 A:middle So we multiply by negative 2. 00:06:36.166 --> 00:06:38.696 A:middle What we do to one side we have to do to the other side 00:06:38.696 --> 00:06:41.516 A:middle so we multiply both sides by negative 2. 00:06:41.756 --> 00:06:44.266 A:middle Since we're multiplying fractions, we'll go ahead 00:06:44.266 --> 00:06:46.926 A:middle and write the negative 2 as negative 2 over 1. 00:06:47.286 --> 00:06:49.686 A:middle And you can see that the negative 2s cancel. 00:06:50.286 --> 00:06:59.846 A:middle So we just get 1x equals negative 2 times 3 which is negative 6 so x equals negative 6. 00:07:03.856 --> 00:07:09.806 A:middle Example. Solve for x. 3x plus 7 equals 5x plus 7. 00:07:10.286 --> 00:07:15.336 A:middle In this equation, we have variables on both sides and numbers on both sides. 00:07:15.716 --> 00:07:20.946 A:middle So we need to get all of the x's on one side and all of the numbers on the other side. 00:07:21.466 --> 00:07:23.796 A:middle So first let's move the x's. 00:07:24.856 --> 00:07:28.196 A:middle I choose to move the x's to the right side. 00:07:28.716 --> 00:07:31.066 A:middle So I'm going to move the 3x. 00:07:31.216 --> 00:07:31.956 A:middle It doesn't matter. 00:07:32.076 --> 00:07:33.266 A:middle You can choose either side. 00:07:33.586 --> 00:07:38.566 A:middle So we would subtract 3x from both sides because that's a positive 3x. 00:07:39.176 --> 00:07:43.206 A:middle When I subtract 3x from positive 3x, I get 0. 00:07:43.916 --> 00:07:46.456 A:middle So I have 0 plus 7 equals. 00:07:46.846 --> 00:07:52.936 A:middle On the right-hand side, I have 5x minus 3x which is 2x and I still have the plus 7. 00:07:52.996 --> 00:07:58.876 A:middle Now I need to move the 7 to the left-hand side because I want to get x by itself. 00:07:59.626 --> 00:08:01.896 A:middle So I subtract 7 from both sides. 00:08:02.226 --> 00:08:05.256 A:middle On the left-hand side, 7 minus 7 is 0. 00:08:05.256 --> 00:08:09.866 A:middle On the right-hand side, I have also 7 minus 7 is 0. 00:08:10.256 --> 00:08:13.826 A:middle So I have 2x plus 0 which is just 2x. 00:08:14.566 --> 00:08:17.646 A:middle I still do not have x by itself. 00:08:17.996 --> 00:08:21.946 A:middle There is a 2 that's being multiplied times the x so I need 00:08:21.946 --> 00:08:24.156 A:middle to do the inverse operation which is divide. 00:08:24.696 --> 00:08:26.486 A:middle So I divide both sides by 2. 00:08:26.906 --> 00:08:29.286 A:middle 0 divided by 2 is 0. 00:08:29.996 --> 00:08:34.016 A:middle And on the right-hand side the 2s cancel so I get 1x. 00:08:34.406 --> 00:08:37.166 A:middle Well, that's just x so x is equal to 0. 00:08:37.586 --> 00:08:39.976 A:middle And you can turn it around and put the x on the other side. 00:08:45.456 --> 00:08:50.296 A:middle Example. Solve for x. 5x minus 8 equals 6x minus 1. 00:08:50.926 --> 00:08:55.316 A:middle Once again we have variables on both sides and numbers on both sides, so we have to choose 00:08:55.316 --> 00:08:57.956 A:middle which side we want to put the x's on. 00:08:57.956 --> 00:08:59.146 A:middle I choose the left-hand side this time. 00:08:59.436 --> 00:09:02.136 A:middle So I'm going to move the 6x to the left-hand side. 00:09:02.576 --> 00:09:05.736 A:middle Since it's a positive 6x, I'm going to subtract 6x. 00:09:06.066 --> 00:09:08.156 A:middle I subtract 6x from both sides. 00:09:08.416 --> 00:09:10.946 A:middle 5x minus 6x is negative 1x. 00:09:11.196 --> 00:09:14.256 A:middle So on the left-hand side I have negative x minus 8. 00:09:14.656 --> 00:09:21.826 A:middle On the right-hand side, 6x minus 6x is 0 and 0 minus 1 is negative 1. 00:09:22.626 --> 00:09:26.046 A:middle Now we reverse our order of operations to isolate the x. 00:09:26.046 --> 00:09:29.246 A:middle So since we have subtract 8, we need to add 8. 00:09:29.796 --> 00:09:31.086 A:middle Add 8 to both sides. 00:09:31.946 --> 00:09:37.706 A:middle Subtract 8 and add 8 gives us 0 so we have negative x plus 0 on the left-hand side. 00:09:37.996 --> 00:09:44.556 A:middle On the right-hand side, we have negative 1 plus 8 and that's 7 so negative x is equal to 7. 00:09:45.436 --> 00:09:48.686 A:middle We still do not have x isolated. 00:09:48.836 --> 00:09:52.076 A:middle We need to get rid of the negative and that's the same thing as a negative 1. 00:09:52.386 --> 00:09:55.286 A:middle So let's divide both sides by negative 1. 00:09:55.736 --> 00:10:01.186 A:middle A negative divided by a negative is a positive so we get 1x on the left-hand side. 00:10:01.976 --> 00:10:05.426 A:middle On the right-hand side, 7 divided by negative 1 is negative 7 00:10:06.026 --> 00:10:09.326 A:middle so we get that x is equal to negative 7. 00:10:09.856 --> 00:10:15.596 A:middle Pause the video and try these problems. 00:10:25.486 --> 00:10:26.266 A:middle Objective three. 00:10:26.506 --> 00:10:30.346 A:middle Solving linear equations that can be simplified by combining like terms. 00:10:31.166 --> 00:10:38.186 A:middle Example. Solve for x. 1 plus 4x plus 2 is equal to negative 3 plus 2x plus 14. 00:10:38.766 --> 00:10:43.526 A:middle So we need to simplify each side of the equation before we start moving things around. 00:10:43.526 --> 00:10:49.126 A:middle So on the left-hand side of the equation we have the like terms of 1 and 2. 00:10:49.766 --> 00:10:51.696 A:middle So 1 plus 2 is equal to 3. 00:10:51.696 --> 00:10:54.766 A:middle So on the left-hand side we have 3 plus 4x. 00:10:55.236 --> 00:10:58.856 A:middle On the right-hand side, we have negative 3 plus 14. 00:10:58.856 --> 00:10:59.916 A:middle Those are like terms. 00:10:59.916 --> 00:11:01.876 A:middle We can add those up and get 11. 00:11:02.386 --> 00:11:05.446 A:middle So the right-hand side is 11 plus 2x. 00:11:06.586 --> 00:11:11.626 A:middle Now we need to move all of the x's to one side and all the numbers to the other side. 00:11:12.086 --> 00:11:14.206 A:middle x's, I choose left side this time. 00:11:15.026 --> 00:11:17.946 A:middle So we're going to move the 2x to the left-hand side 00:11:18.276 --> 00:11:20.916 A:middle which means we need to subtract 2x from both sides. 00:11:21.486 --> 00:11:25.226 A:middle 4x minus 2x is equal to positive 2x. 00:11:25.816 --> 00:11:29.086 A:middle On the left-hand side, we have 3 plus 2x. 00:11:29.436 --> 00:11:35.186 A:middle On the right-hand side, 2x minus 2x is 0 and 11 plus 0 is just 11. 00:11:36.436 --> 00:11:39.456 A:middle Now we reverse our order of operations to isolate the x. 00:11:40.836 --> 00:11:43.956 A:middle So we subtract 3 first before we divide by the 2. 00:11:44.796 --> 00:11:47.306 A:middle So we subtract 3 from both sides of the equation. 00:11:47.766 --> 00:11:52.656 A:middle 3 minus 3 is equal to 0 and we get 0 plus 2x which is just 2x. 00:11:52.946 --> 00:11:57.646 A:middle On the right-hand side, we have 11 minus 3 and that's 8. 00:11:57.896 --> 00:12:00.356 A:middle Now we have 2 times x is equal to 8. 00:12:00.696 --> 00:12:05.316 A:middle We need to isolate the x. The inverse operation to multiply is divide. 00:12:05.626 --> 00:12:07.456 A:middle So we divide both sides by 2. 00:12:07.846 --> 00:12:12.206 A:middle See how the 2s cancel, and on the left-hand side I get 1x. 00:12:12.706 --> 00:12:16.696 A:middle So we get that x is equal to 8 divided by 2 which is 4. 00:12:22.336 --> 00:12:29.276 A:middle Example. Solve for x. Negative 2 times the quantity 3x minus 4 is equal to 2x. 00:12:29.886 --> 00:12:32.236 A:middle Any time you have parentheses in an equation, 00:12:32.626 --> 00:12:36.096 A:middle you must get rid of the parentheses before you start moving things around. 00:12:37.046 --> 00:12:42.476 A:middle So we will distribute the negative 2 to the 3x which gives us negative 6x. 00:12:43.126 --> 00:12:46.036 A:middle We distribute the negative 2 to the negative 4. 00:12:46.406 --> 00:12:48.206 A:middle That gives us positive 8. 00:12:48.856 --> 00:12:52.206 A:middle So the left-hand side is negative 6x plus 8. 00:12:52.476 --> 00:12:54.486 A:middle The right-hand side is just 2x. 00:12:55.256 --> 00:12:59.146 A:middle Now you can move the variables to the left or to the right 00:12:59.966 --> 00:13:07.116 A:middle but since the right-hand side only has the variable 2x, it would be quicker to move all 00:13:07.116 --> 00:13:08.916 A:middle of the x's to the right-hand side. 00:13:09.506 --> 00:13:14.416 A:middle So I'll add 6x to both sides because I had a negative 6x. 00:13:14.816 --> 00:13:21.416 A:middle Negative 6x plus 6x is equal to 0 so the left-hand side is just 8. 00:13:21.766 --> 00:13:24.666 A:middle 2x plus 6x is 8x. 00:13:25.106 --> 00:13:30.366 A:middle 8 is being multiplied times x so the next step is to divide both sides by 8. 00:13:30.766 --> 00:13:32.046 A:middle The 8s cancel. 00:13:32.346 --> 00:13:37.016 A:middle On the left-hand side I get 1 and the right-hand side I get 1x 00:13:37.016 --> 00:13:42.556 A:middle which is just x. You can write the x in the answer either way on the left or the right 00:13:42.556 --> 00:13:45.416 A:middle so 1 equals x is the same as x equals 1. 00:13:50.956 --> 00:13:54.926 A:middle Example. Solve for k. The opposite 00:13:55.146 --> 00:14:05.386 A:middle of the quantity 6k minus 5 minus the quantity negative 5k plus 8 is equal to negative 3. 00:14:06.136 --> 00:14:11.136 A:middle So since we have parentheses, our first goal is to get rid of the parentheses. 00:14:11.536 --> 00:14:17.036 A:middle A negative in front of a parentheses is the same thing as negative 1 times the parentheses. 00:14:17.306 --> 00:14:21.396 A:middle So we're going to distribute a negative 1 to each of these parentheses. 00:14:22.386 --> 00:14:28.866 A:middle The 6k inside the first parentheses will become a negative 6k when we multiply it by negative 1. 00:14:29.486 --> 00:14:31.026 A:middle Subtract 5. 00:14:31.056 --> 00:14:34.176 A:middle Well, that will become add 5 when we distribute the negative 1. 00:14:34.736 --> 00:14:39.316 A:middle And negative 1 times negative 5k gives us positive 5k. 00:14:40.016 --> 00:14:43.556 A:middle Negative 1 times positive 8 gives us negative 8. 00:14:44.256 --> 00:14:47.846 A:middle And now we have the parentheses all gone. 00:14:48.176 --> 00:14:49.886 A:middle Now we're going to have like terms. 00:14:50.116 --> 00:14:56.066 A:middle On the left-hand side, we have a negative 6k and a 5k which adds to a negative 1k. 00:14:56.566 --> 00:14:59.806 A:middle And we also have 5 minus 8. 00:15:00.516 --> 00:15:02.016 A:middle Those add up to negative 3. 00:15:04.086 --> 00:15:06.856 A:middle Now we wish to get k by itself. 00:15:07.286 --> 00:15:12.126 A:middle So the first step would be to add 3 to both sides because we have a minus 3. 00:15:13.346 --> 00:15:16.986 A:middle Minus 3 plus 3 is equal to 0. 00:15:17.166 --> 00:15:19.836 A:middle So we get negative k on the left-hand side. 00:15:20.396 --> 00:15:24.456 A:middle On the right-hand side, negative 3 plus 3 is equal to 0. 00:15:25.406 --> 00:15:27.456 A:middle We still do not have the k isolated. 00:15:27.816 --> 00:15:29.186 A:middle We need to get rid of the negative. 00:15:29.836 --> 00:15:31.966 A:middle That's the same thing as negative 1k. 00:15:32.536 --> 00:15:35.786 A:middle So we're going to divide both sides by negative 1. 00:15:36.516 --> 00:15:38.806 A:middle Negative k divided by negative 1. 00:15:39.006 --> 00:15:42.836 A:middle Negative divided by a negative is a positive so we just get k on the left-hand side. 00:15:43.896 --> 00:15:46.416 A:middle 0 divided by negative 1 is 0. 00:15:47.266 --> 00:15:48.916 A:middle Therefore, k is equal to 0. 00:15:53.836 --> 00:16:02.506 A:middle Example. Solve for m. Negative 3 times the quantity m minus 4 plus 2 times the quantity 5 00:16:02.576 --> 00:16:05.196 A:middle plus 2m is equal to 29. 00:16:05.956 --> 00:16:08.736 A:middle So our first goal is to get rid of the parentheses. 00:16:09.486 --> 00:16:13.346 A:middle We distribute the negative 3 to m and get negative 3m. 00:16:13.916 --> 00:16:17.716 A:middle Negative 3 times negative 4 gives us plus 12. 00:16:18.526 --> 00:16:21.526 A:middle 2 times 5 gives us 10. 00:16:22.136 --> 00:16:25.126 A:middle And 2 times 2m gives us 4m. 00:16:25.986 --> 00:16:29.636 A:middle Now we need to add like terms on the left-hand side of the equation. 00:16:30.656 --> 00:16:35.216 A:middle Negative 3m and a positive 4m gives us positive 1m. 00:16:36.476 --> 00:16:40.716 A:middle Positive 12 and positive 10 give us 22. 00:16:41.606 --> 00:16:45.276 A:middle Now we have m plus 22 equals 29. 00:16:46.126 --> 00:16:50.526 A:middle To get the m by itself, we need to subtract 22 from both sides. 00:16:51.246 --> 00:16:58.606 A:middle So we get m equals 29 minus 22 which is 7 so m is equal to 7. 00:16:58.926 --> 00:17:04.946 A:middle Pause the video and try these problems.