now what can we do to solve a system of three equations let's say that two x plus y plus z is equal to seven and two x minus y plus two z is equal to six and x minus two y plus z is equal to zero feel free to pause the video if you know what to do if you know how to solve it so what we need to do is choose two out of the three equations let's choose the first two equations because if we add them notice that we can cancel y so first i'm going to rewrite it so once we add them 2x plus 2x is 4x z plus 2z is 3z and 7 plus 6 is 13. now i'm going to save this equation now what we need to do next is we need to combine another two equations and cancel the same variable y but we need to include equation three because we didn't use it we only use the first two equations we can use equation one and three or two and three let's use one and three to cancel y we need to multiply the first equation by two so two x times two is four x y times two is two y z times 2 is 2z 7 times 2 is 14. now let's rewrite the third equation right beneath the modified first equation and let's add them 2y and negative 2y cancels 4x plus x is 5x 2z plus z is 3z so here's the second equation that we have in terms of x and z 5x plus 3z is equal to 14. now to solve those two equations we just need to multiply one of them by negative one let's multiply the first equation by minus one so it's going to be negative four x minus 3z which is equal to negative 13. and let's rewrite the other equation and right beneath it so let's add the two equations so we can see that z will cancel negative 4x plus 5x is 1x negative 13 plus 14 is 1. so therefore x is equal to 1. now let's plug that value into the first equation we're not the first equation but the second equation in terms of x and z so it's going to be five times one plus three z is equal to fourteen if we subtract both sides by five fourteen minus five is nine and then we'll divide both sides by three nine divided by three is three so z is equal to three now we need to find the value of y let's use the first equation in its unmodified state the original first equation so x is one we don't know the value of y z is three so let's combine like terms two plus three is five so y plus five is seven now let's subtract both sides by five seven minus five is two so y is equal to two so we can write the answer like this one comma two comma three x y z so that's it for that problem now let's work on some word problems two investments totaling thirteen thousand were placed in separate accounts earning fifteen percent and fourteen percent annually if the total interest received was nineteen hundred during the first year how much money was invested in the account paying 15 interest well let's write an equation let's say that x is the amount of money paid or placed in the first account and y is for the other count x plus y has to add up to the total investment of thirteen thousand now the total interest is nineteen hundred so the first account which x amount of money was placed in let's say that account paid fifteen percent interest fifteen percent as the decimal is point one five to convert a percentage into a decimal divide by a hundred fourteen divided by a hundred is point one four so this should equal 1900 the total interest received from both accounts so now that we have a system of equations we can solve it whenever you have two variables you need two equations to solve those two variables and since we have decimals let's multiply the second equation by a hundred point fifteen times a hundred is fifteen and point fourteen times a hundred is fourteen nineteen hundred times a hundred is a hundred ninety thousand just add two zeros now let's solve this by elimination let's cancel the y variable so let's multiply the first equation by negative 14. so we're going to have negative 14x minus 14y and then 13000 times negative 14 is negative 182 000. now let's add the two equations 15 minus 14 is x the y's will cancel and 190 000 minus 182 000 is 8 000. so that's how much money was invested in the account paying 15 interest because x is associated with 15 now the total investment is 13 000 which means that 5 000 was placed in the other account so that's the account that was paying 14 interest and so that's it for this problem now let's make sense of that problem so 8 000 was placed in the first account which paid 15 percent interest fifteen percent of eight thousand if you multiply these two numbers is twelve hundred so the first account earned twelve hundred in interest alone now the second account received five thousand dollars and was paying 14 interest so 5 000 times 0.14 is 700. so that account received 700 in interest for that year the total interest is 1200 plus seven hundred which is nineteen hundred so as you can see the numbers make sense if five apples and eight bananas cost six dollars and fifty five cents and if nine apples and seven bananas cost nine dollars and twenty cents what is the cost of seven apples and ten bananas we have two variables apples and bananas we'll use a and b to represent those variables so we need to write two equations if we could find the value of one apple and one banana then we could find a value of seven apples and ten bananas which is the goal of the problem so the first part of the problem states that 5 apples or 5a plus 8 bananas has a cost of 6.55 cents so 5a plus 8b equals six point fifty five nine apples and seven bananas cost nine dollars and twenty cents let's use these two equations to find the value of each apple and each banana so what do you think we should do now if you want to we can get rid of the decimal we can multiply both sides by 100 but we don't have to let's multiply the first equation by a negative seven to get rid of b and let's multiply the second equation by positive eight we want to get 56b 5a times negative 7 is going to be negative 35a 8b times negative 7. is negative 56b and 6.55 times negative seven that's going to be negative 45.85 9a times 8 is 72a 7b times 8 is positive 56b 9.20 times 8 is 73.60 now let's add the two equations negative 35 plus 72 that's going to be positive 37 or 37a negative 45.85 plus 73.60 that's 27.75 now a is going to be 27.75 divided by 37. so the cost of each apple is 75 cents now let's use the first equation in its unmodified form to find b so 5 times 0.75 plus 8b is equal to 6.55 so 5 times 0.75 is 3.75 now let's subtract both sides by 3.75 so 6.55 minus 3.75 and that's equal to 2.8 and b is going to be 2.8 divided by 8 which is 35 cents so now we have the cost of every apple and every banana so the cost of a single apple is 75 cents and the cost of a single banana is 35 cents so now we can find the value of seven apples and 10 bananas so it's going to be 7 times 75 cents plus 10 times 35 cents 7 times 0.75 is 5. and 25 cents 10 times 0.35 you just got to move the decimal that's three dollars and fifty cents so if we add this this should be eight seventy five so that's the cost of seven apples and ten bananas it's eight dollars and seventy five cents you