[Music] hello and welcome to algebra 1 by emath instruction my name is kirk wyler and today we're starting a brand new course here algebra one unit one lesson one on variables and expressions and i'm excited to start this course with you right we've got a long road ahead of us but by the end of it all you'll be much better at algebra than you were in middle school and hopefully you'll have the building blocks for success in future math courses so without further ado let's launch right into our lesson on variables and expressions all right so first let's talk about what an expression is an expression is any combination of numbers and or variables and using addition multiplication subtraction and or other operations such as exponents right so and this is important in math you know you've got numbers that just kind of hang out right but then you've got expressions specifically numbers and variables that are combined with addition subtraction multiplication division roots exponents etc right so let's jump right in and look at just some numerical expressions and how to evaluate them let's take a look at exercise number one review the order of operations by giving the value of each of the following purely numerical expressions do these without a calculator in order to review basic middle school number concepts all right awesome so let me just bring this up to the top of the sheet so we've got a little bit more room to work with now keep in mind order of operations are our rules for the order in which we are supposed to evaluate or interpret algebraic expressions so let's take a look at letter a right in letter a we have the expression three times two plus seven and in this case we're showing the multiplication with the little dot symbol now our order of operations tells us that multiplication must come before addition so in other words i want to think about this and evaluate it by saying well the first thing i'm going to do is do 3 times 2 which is just 6 and then i'm going to evaluate 6 plus 7 and i'm going to get 13 right and that is exceptionally important because if we look at this and interpret it as well i'm going to add the 2 and the 7 first which would give me 9 and then multiply by 3 which would give me 27 i'm going to get an entirely different numerical result that's where this order of operations is so key we have to know how to do it alright so what i'd like you to do right now is i'd like you to pause the video and i'd like you to just attempt letter b and letter c see if you can remember your order of operations and how to do those and then we'll move on and together we'll take a look at the second row of problems that are a little bit more challenging all right now again generally speaking when we read an expression we want to read it from left to right so in letter b we have 8 minus 1 half times 6 plus 24 divided by 6. now again order of operations tells us that we should be doing the multiplication and the division first which one of them should we do first well multiplication and division tie with one another and so when you have a tie in order of operations you really kind of do it just from left to right so what i'm going to be doing is i'm going to be figuring out what one half times 6 is and what 24 divided by 6 is and i'm going to be putting those into the expression now keep in mind right that one half times 6 is the same as 6 divided by 2 so that's 3. 24 divided by 6 is equal to 4 and now that i have only subtraction and addition i really do it from left to right so i have 8 minus 3 which is 5 plus 4 and 5 plus 4 gives me a final numerical result of 9. all right order of operations also tells us that when we have quantities within parentheses those quantities should be evaluated first before we do anything else so in letter c we have 4 times 8 minus 6 minus 7 times 5 minus 3. so the first thing i'm going to do is those very easy differences in the parentheses right and i'm going to get 4 times 8 minus 6 which is 2 minus 7 times 5 minus three which is also two now i'm going to do the two multiplications so four minus two four times two is eight seven times two is fourteen and here's where you wanna be very careful right whenever i have a smaller number minus a larger number i get a negative and specifically in this case i get negative six right so we do those operations that are within the parentheses first and then we proceed afterwards now let's take a look at some more challenging problems in d e and f all right so looking at letter d we finally have some exponents involved right i've got five to the second four to the second plus three i've got a fraction involved whenever i have a fraction involved we really want to look at the numerator the top and the denominator the bottom as if they have parentheses around them right i want to evaluate that entire numerator that entire denominator and then do the division or do the simplification of the fraction that's sitting there so when i look at this numerator and i see 5 to the second minus 4 to the second plus 3 i want to do those exponents first all right so i get 5 to the second that's five times five minus four to the second that's four times four and plus three and i can evaluate the denominator one minus five is negative four now i can evaluate that numerator going from left to right 25 minus 16 is 9 right so i have 9 plus 3 9 plus 3 is 12 and now i have 12 divided by negative 4. keep in mind that when you have a positive divided by a negative you get a negative and so i have a final evaluation of negative three all right exponents come before multiplication and division they come after parentheses though so if you had parentheses those would take precedent take a look at letter c here we finally have a square root involved we have 22 minus the square root of 36 plus 64. and just like numerators of fractions we want to kind of think about there being parentheses within this quantity under the square root so we want to do that addition first then we want to find the square root and after we do that we want to do the subtraction pause the video now and see if you can work on both letter e and letter f try to get final answers and then we'll go through both of them take a couple minutes all right let's do it as i said in letter e the first thing i want to do is i want to add the 36 and the 64 right so when i do that 36 plus 64 is 100 right now i want to think about what the square root of 100 is remember the square root is the number you have to multiply by itself to get the number under the square root so that's going to be 10 right the square root of 100 10 times 10. let me do this and of course 22 minus 10 is just 12. that's easy enough all right now let's look at one that looks absolutely horrible right because i've got negative 16 divided by two plus five times two all divided by two to the third another exponent going on there there's a lot of different things i can do keep in mind of course the numerator and the denominator are completely separate from each other i really only want to have those two interact kind of like what i did in letter d once i've simplified everything so let's jump right into this problem all right in the numerator i can do that division and multiplication negative 16 divided by two is going to be negative eight five times two is ten now i can do the exponent the denominator two to the third remember is two times two times two and that's equal to eight right i can now simplify the numerator negative eight plus ten is a positive two and two eighths of course i can simplify this fraction by dividing the numerator and denominator by two and i find the final answer there is one fourth all right so we're just working through orders of operations remember parentheses exponents multiplication and division tie and then addition and subtraction and they also tie and you kind of break the tie by just moving from left to right now all of these expressions were completely numerical let's get into some that involve some variables all right so once an expression involves a variable it's important to be able to read the expression be able to look at it and sort of understand what's being done to the variable without having to like plug a number in so let's get a little bit of practice on that with exercise number two if the letter x represents some unknown quantity explain the calculations slash operations that each of the following expressions represent state both the operations and the order in which they occur all right so in other words when i look at this expression 3x minus 8 that's the way i would read it i want to immediately know what's happening to x now again i do this by thinking about my order of operations right if i took a value of x and i actually replaced the variable with that value of x the question would be well what would what would happen right well order of operations says the first thing i'm going to do is the multiplication remember that 3 just sitting beside the x means that we're doing 3 times x so the first thing right is where multi-multiplying by three right that's the first thing and then we're taking the result that looks beautiful taking the result and subtracting eight my handwriting's gonna eventually get better i just kind of a little out of practice now that we're in a new course right but i wanna be able to look at this and immediately know oh what that expression is doing is taking that value of x whatever it is we don't know at this point multiplying it by 3 and then taking that result and subtracting 8 from it what i am most certainly not doing is taking x subtracting 8 from it and then multiplying that result by 3. three alright so see if you can pause the video now and write down what's happening in both letter b and letter c to that variable x all right well letter b is fairly simple letter c is a little bit more complex right in letter b right what i've got the first thing that's happening is i'm subtracting 4 from x the second thing i'm doing is i'm taking the result and i'm dividing it by two remember fractions are division you're always taking the numerator and you're dividing it by the denominator so in letter b give myself a little bit of room first thing that we're doing is we're subtracting subtracting four and the second thing is we're dividing by two ah my mystery red pen finally made an appearance right now letter c again it's important when we look at four times x squared minus eight right we have to know that the first thing that's happening is we're squaring that x right we're multiplying it by itself then we're doing the multiplication by four and then we're subtracting eight right so these three different steps the first one being we are squaring x then we're multiplying by four and then we are subtracting eight and it's critical because so much of algebra involves expressions that have variables in them if you are reading them incorrectly if you're thinking about the order that they're done in in in the incorrect order then it's going to be chaos right it's just going to be pure chaos all right let's keep going now of course what we want to be able to do is sort of a combination of what we did in exercise one and what we did in exercise two which is evaluating expressions now what does it mean to evaluate expression right it's finding the value of an expression containing one or more variables when the values of those variables are known right so we have to we have to actually know what the value of the variable is but once we do we have to feel confident with our order of operations and just our ability with numbers and stuff to be able to actually come up with the value of an algebraic expression if we know what the variables are equal to so let's take a look at some of that in exercise number three find the value of each expression given show the steps you use to find your final answer only use your calculator if necessary to check your final results simplify any fractional answers all right great this truly is a combination of what we did in exercise one and what we did in exercise two right so if i look at letter a i've got the expression four x minus seven when x is equal to 5. in other words i want to know what 4x minus 7 is when x is 5. so the first thing i have to know that what's going on here is i'm taking x i'm multiplying it by 4 and then i'm subtracting 7. now i want to substitute in the value of the variable so what does that look like right so i want to be careful i like to show multiplication by using a parentheses right so i've got 4 times minus seven i now do four times five which is twenty minus seven and twenty minus seven is thirteen so the expression four x minus seven has lots of different values depending on the value of x but when x is equal to 5 that particular expression is equal to 13. all right let's do another one of them together and then have you do some of the more challenging ones on your own and then we'll go through them right so in letter b we've got 25 minus two-thirds times n and again keeping in mind what we're really doing is we're taking the value of n which in this case is going to be 12 we're multiplying it by two-thirds and then we're subtracting that result from 25. so let's take a look after we do that substitution right 25 minus two-thirds times 12. now keep in mind whenever you multiply by a fraction you're really doing a division problem 3 goes into 12 4 times and then you're doing a multiplication problem 2 times 4 is 8 right and now of course 25 minus 8 is 17. you can of course use your calculators to do many of these things but we try to keep the numbers purposely small and relatively easy to work with early on in this course because we want you to build up numerical fluency without the calculator so try as many of these as possible without using your calculator but certainly if the numerical work is getting in your way grab that calculator that's what it's there for all right let's take a look at some that are a little bit more challenging all right now keep in mind right order of operations parentheses exponents multiplication and division tie and then addition subtraction tie and that's important because here we've got some exponents going on in letter c all right let's do letter c together and then we'll have you do letter d on your own so letter c we've got c squared c to the second plus 6 times c for when c is equal to negative 4. now i want to be good about my my substitution work here because i've got a negative number going on all right it's great to put that negative number in parentheses so i have negative 4 squared plus 6 times negative 4. keep in mind that negative 4 squared negative 4 times negative 4 is positive 16. then a positive 6 times a negative 4 is going to be negative 24. all right so i now have positive 16 plus negative 24 is going to leave me with an overall negative eight all right again easy enough if you know your order of operations and you know some basic facts about negative and positive numbers why don't you go ahead and try letter d pause the video now and take a few minutes to try that one all right let's go through it so in letter d we have one half times t squared minus 2 times t plus 9 for t equals 6. again let's be careful right when i do the substitution i'll almost always put whatever value i'm substituting into parentheses all right order of operations tells me i should do that 6 squared first 6 times 6 is 36. maybe we'll leave all the other multiplication alone for right now now one half times 36 remember that's just 36 divided by 2 so that's 18. 2 times 6 is 12 plus 9. now i'm just going to add from left to right 18 minus 12 is 6 6 plus 9 is 15. all right great i hope you got that one right we've got one more row to go on these evaluations so let's take a look at letter e and letter f all right letter e we've got the square root of five x minus four and again keep in mind things like square roots and denominators of fractions you can really think of these as having parentheses around them right they're just sort of like naturally grouped so i want to figure out what that 5x minus 4 is when x is equal to 8 and then i want to take the square root of it so pause the video now and see if you can actually do both letter e and letter f see what you can do on those all right let's do it so letter e right first thing i'm going to do is put the square root make it nice and big 5 times 8 minus 4. i am literally going to just ignore the square root symbol until i have a single number underneath it 5 times 8 is 40 minus 4 still ignoring the square root 40 minus 4 is 36 and finally i can say all right what is the square root of 36 well that's 6 right because 6 times 6 is 36 all right but keep in mind right i didn't even bring that square root in until the absolute last moment very similar to the fraction right here i've got 6 divided by y cubed plus 4 for y is equal to 2. so first things first i'm gonna put the 2 in cube it plus 4 right i'm going to ignore that numerator nothing i can do with that i'm going to do 2 to the third 2 times 2 times 2 is 8 plus 4. now i'm going to sum in the denominator 8 plus 4 is 12 and i can divide both numerator and denominator by six and i find that my final result there is one half all right just working it down right working it down using your order of operations all right let's do a little bit more one more problem here we go exercise number four the amount of money in dollars a tour company earns on a tour can be calculated using the expression 12 times parentheses n minus 4 where n is the number of people who go on the tour how much money do they make when 14 people go on the tour all right so there's a lot of words here that you got to kind of work through but everything that you need to figure out the answer to this problem is sitting right in there pause the video now and see if you can figure out how much money the tour company makes when 14 people go on tour all right well we are told that this expression 12 times n minus 4 let me just circle it right is going to calculate what we want to know how much money they made by going on the tour company by by taking this tour right and the n actually represents something now n is the number of people who go on tour so how much money do they make when 14 people go on tour well what that tells us is it tells us that n is equal to 14 and the amount of money that they're going to make is going to be 12 times 14 minus 4 right order of operations tells us that we've got to do that what's in parentheses first 14 minus 4 is 10 and now 12 times 10 is 120. so they must make 120 when 14 people go on tour right that's easy simple to do something like that but it goes to to illustrate a point which is very often in real world problems right the variables stand for something in this particular problem n stood for the number of people that went on tour and the expression itself the 12 times n minus four right that represented how much money we would make given that a certain number of people went on tour so we could use that expression to figure out what we were looking for all right let's wrap this up alright so today we got into some real basics stuff that you probably almost rolled your eyes at because especially the numerical work you were probably doing since you were in middle school right maybe even as early as like sixth or seventh grade right even the variable work you certainly have had exposure to in previous courses but it's important because variables and algebraic expressions in other words expressions that involve variables are going to become the basis of so much of what we do and so much of what we are we do is going to boil down to can you look at an expression understand what is going on to the variable or variables in the expression can you evaluate the expression if you know values of the variables right and all that's going to become important especially when we then start to undo what's been done to the variables in order to solve equations all right but more on that in future lessons for now i just want to thank you for joining me for another algebra 1 lesson by emath instruction my name is kirk weiler and until next time keep thinking and keep solving problems