[Music] in this video we will solve problems involving systems of linear equations in two variables so our objective we will solve problems involving systems of linear equations in two variables ayun okay so let's have example number one the sum of two numbers is 10 and their difference is 46. what are the numbers so first step you have to understand the problem so highlights newport yuma important so we have the sum of two numbers is 10 and the difference is 46 okay so ebik's a b hen numbers their difference is 46. so let us represent the given a problem so therefore your x will be your larger number and then your y is the smaller number okay so your x is your larger number and your y is the smaller number so let us plan so let's plan out and then let us represent the given equation so your equation one will be since the sum of two numbers is ten so therefore x plus y is equal to ten and their difference is forty-six therefore that is x minus y is equal to forty-six okay so this will be our system now what do you think is um the method that is more convenient to use so given the system and the genuine mass modeling okay obviously it's elimination method why not additive inverse okay so we will be using the elimination method so let us carry out the plan so i have x plus y is equal to 10 and x minus y is equal to 46. so what you're going to do is to cancel it on y terms so we now have 2x is equal to 56 divide both sides by 2 this will be cancelled so your x is equal to 28 okay now after getting the value of 20 at x the value of x then you can get any equation original equation so let us use x plus y is equal to 10 okay and then substitute x so we will substitute 28 x so we now have 28 plus y that is equal to 10 so y is equal to 10 minus 28 so my punta shadito therefore our y is equal to negative 18. okay now first step is you have to understand the problem step two is you have to plan out and then step three carry out the plan after you got your x and y values step four you have to check the result okay so the solution is 28 negative 18 or x is equal to 2 28 and y is equal to negative 18 so it upon so what you're going to do is you have to check okay not in your equation 1 10 which is x plus y is equal to 10 and then your equation to nothing which is x minus 1 is equal to 46 so substitute values equals so our x is 28 and our y is negative eighteen so ilala okay so 28 plus negative 18 so different signs it all you have to subtract and that is positive 10 so as you can see they are equal and this is checked okay so therefore it satisfies the given equation densa system nathaniel first equation next procedure is equation so 28 minus your y is negative 18. so you have to enclose this um using parenthesis so that will become 28 plus because a negative times negative positive so 28 plus 18 that is equal to 46 so as you can see they are equal so this is check okay so since it satisfies both equations therefore the solution is 28 negative 18. so therefore the two numbers are 28 and negative 18. let's have example two a carpenter wants to cut a 16 foot piece of wood into two pieces the longer piece is to be one foot longer than twice the shorter piece find the length of each piece okay so first step understand the problem highlights nothing among important data so we have 16 foot piece of wood into two pieces and then the longer pieces to be one foot longer than twice the shorter piece so that pad is you have to represent the unknown young longer at young shorter pismo so let us represent the x as the length of the shorter piece and then your y as the length of the longer piece okay so that you know so you can represent okay now so we now have the representation of the shorter and longer piece then what will be the next um step so we have um the longer piece so make the translate the longer piece down so longer piece is equal to y so that a new representation so the longer piece is um at the longer piece most of why is equal to is to be one foot longer so ebiksa being one foot longer twice the shorter piece shorter piece is represented as x so if it's a being twice x okay so the longer piece is equal to is to be one foot longer than twice the shorter piece since ethereum representation not in a shorter piece now so let us represent the equation we now have y is equal to twice is it is a meaning two okay so two x plus one now okay so let us plan out so your equation one will be so balika not ends then what will be your what's supposed to be your um equation one so but they cannot even problem a carpenter wants to cut a 16 foot piece of wood into two pieces so you 16 foot into two pieces so ebics have been entering two pieces so young two pieces that is equivalent to sixteen feet okay so x plus y is equal to sixteen foot piece of wood into two pieces therefore it in two pieces that is that must be equivalent to 16. okay so that will be your first equation and then your second equation and that is y is equal to two x plus one so this will be our systems of linear equation from the given problem now so as you can see we already have here the equation of solving for x in terms of solving for y in terms of x so no none has solved for y in terms of x so therefore equations okay that is substitution method so let us continue so this is your first equation and then this is your second equation so let us solve gamma tang equation one so we have x plus y is equal to 16 so since i'm valuing y not in d the y is equal to two x plus one so it is a substitute not indeed so y i am four so ill again say two x plus one then so y and then so we now have x so plus two x plus one okay equals 16. so we can now combine similar terms so x plus 2x so x plus 2 that is 3x plus 1 is equal to 16. okay so we now have three x is equal to sixteen mapu punta tu dito so i'm getting minus one so sixteen minus one that is fifteen so for us to solve for x divide both sides by three and then this will be cancelled and then x is equal to 5 because 15 divided by that is 5. okay next so let us solve for y so by using uh the second equation so it nut and see five density x so two times five plus one so two times five that is ten plus one that is eleven okay so do not forget to check so we have the solution is 5 11 or x is equal to 5 y is equal to 11. so using the original equation x plus y is equal to 16 and your equation 2 y is equal to 2x plus 1. so we now have substitute x at ymo so five dito eleven details so five plus eleven that is sixteen so as you can see they are equal and this is check okay let's proceed to the other equation so y is equal to two x plus one substitute not in c eleven then some y and then c five density x so so let us solve so we now have eleven is equal to two times 5 that is 10 plus 1 so 10 plus 1 is equal to 11 so as you can see they are also equal so therefore um the solution 511 satisfies equation 1 and equation two so the length of the shorter and longer piece should be five feet and eleven feet respectively so ebik's a b hand your shorter piece is five feet and the longer piece is eleven feet so a check nothing young longer piece though is one foot longer than twice the shorter piece so pack the knife's doughnut in it two times to el ten and one foot longer eleven so ion push okay next let's proceed as a third example alex saved 20 coins consisting of 25 centavo and one peso coins how many of each kind did he have if the coins amounted to 10 25 pesos okay so understand the problem highlights nothing young even meron shang 20 coins it consists of 25 centavo and one peso coin okay so 20. 25 centavo and one and the total amount is 10 25 pesos so let us represent first as 25 centavo and the one peso so let us represent the number young b langnong 25 centavo coins as x and then b log num one peso coins as y okay since we are looking for how many so okay so plan out so what will be your first equation okay first equation actually you first equation equation um so therefore your equation one is x plus y is equal to 20. 25 centavo one peso coin is equivalent to 20 coins so this will be your first equation okay and then your second equation so your second equation is angsa um 25 centavo and one peso coin and the total shannon 1025 so on 25 centavo is equivalent to 0.25 right okay so 0.25 times 25 centavo that's why it's 0.25 x casita times nothing shady so second equation okay now so this will be our system now what do you think is the best or convenient method to be used use substitution puederin you may solve for widely negative x or you may solve for forex magnitude will be using elimination so i will make it negative okay negativity by negative 1 both sides so when we are multiplying a number do answer adding equation always multiply it don't adding terms all of the terms okay so negative 1 times x that is negative x and then negative 1 times y that is negative y and then 20 times negative 1 negative 20. so this is now our equation three okay and then just copy the second equation so therefore we can now cancel y okay one to making 1025 okay and okay one since one thousand and then now so negative 20 plus 10.25 sodium meron so seventy five centimeters so why negative because absolute value okay so for us to cancel um negative zero point seventy five of course we will divide it by negative 0.75 so this will be cancelled okay so we only have now x is equal to divide not in it 75 centigrade centavosa 9 pesos and 75 centavos okay so next so negative x minus y is equal to negative 20 or predicted original equation x plus y is equal to um 20 so negative x minus y is equal to negative 20. substitute nothing say x that will become negative 13 so it will then my negative and then your x minus 13 get negative 13. minus y is equal to negative 20. so we now have negative y is equal to excuse me negative 20 so i'm a gigan positive eto plus 13. so y is equal to negative y is equal to negative 20 plus 13 that is negative seven so since negative it also um i multiply nothing to both sides by negative one to make this positive so my gig in y is equal equal to 7 okay so check the result so we have um 13 7 or x is equal to 13 and y is equal to 7 so this is our solution so let us check going back to the original equation so our original equations are x plus y is equal to 20 and 0.25 x plus y is equal to 10.2 okay so i check not n so your x is 13 and then your y is 7. so 13 plus 7 that is 20 so they are equal and this is check okay next so naman substitute natanzi x um as a thirteen duns i think x and then c seven then so y so 13 times point 25 so 13 25 centavos okay so that is 3.75 okay so therefore i know it should be okay so it should be 3.25 okay um so that is 33.25 centavos okay plus seven so it had nothing to three plus seven that is 10.25 so as you can see they are equal and this is check okay so therefore uh since it satisfies both equations therefore the solution is correct the 13 7. so alex has 13 pieces of 25 centavos and seven pieces of one peso coin um uh thank you for watching this video i hope you learned something don't forget to like subscribe and hit the bell button put updated ko for more video tutorial this is your guide in learning your mod lesson your walmart channel