Germany Can you solve this Math Olympiad X Y

Hello. You're welcome to solve this math problem, which is + is equal to 24. * is equal to 64. To find the values of and y's from these two systems of equations. Now, in the first step, let's start by letting this here as equation one and this as equation two. Then, from equation one, which is this, plus is equal to 24. Into here, we make the subject, so we take to the left side. So, it will be is equal to 24. Take to this side, it will be minus Then, note this equation in terms of Then, in the next step, from equation two, what is equation two? It is * is equal to 64. Now, into we substitute 24 minus So, it will be * which we substitute here, which is 24 minus Bracket is equal to 64. Then, here it will be * 24, it is 24 * negative is negative squared is equal to 64. Then, we start by this negative squared. Then, plus 24 We take 64 to this side, it will be minus 64 is equal to zero. Then, into here, we make this negative squared, we make this positive. So, then we divide the whole equation divide by negative one. So, it will be negative squared divide by negative one is positive squared. Then, 24 divide by negative is minus 24 -64 / -1 is positive 64 is equal to 0 / -1 it is 0. Then from this quadratic equation, we solve this by using quadratic formula. Now by applying quadratic formula to find the values of is equal to -b plus or minus square root of squared minus 4ac over 2a. So here it will be is equal to minus is coefficient of which is -24. Then plus or minus square root of squared it will be -24 bracket squared. Then minus 4 * it is 1 * is 64. Then over 2 * it is 1. Then into here it will be is equal to negative negative negative and negative 24 is positive 24 plus or minus square root of -24 squared it will be negative squared is positive. Now 24 squared is 576. Then minus 4 * 4 it is 16 with 1 4 * 6 is 24. 24 plus 1 is 25. Then over this and this is 2. Then into here it will be is equal to 24 plus or minus square root of we take this minus this Now, 6 - 6 is 0. 7 - 5 it is 2. 5 - 2 it is 3. Then over 2. So, into here it will be is equal to 24 plus or minus square root of 320. Now, here it will be square root. Let's find the prime factors of 320. Now, from 320 here divide by 2 160. Then divide by 2 80. Then divide by 2 40. Then divide by 2 20. Then divide by 2 10. Then divide by 2 5. Then divide by 5 1. Now, here it will be 2 * How many 2s? 1 2 3 4 5 6. So, here it is 2 ^ 6 then times this over this 2. Then in the next step it will be is equal to 24 plus or minus we separate the square root. So, it will be square root of 2 ^ 6 times square root of 5 over 2. Then here it will be is equal to 24 plus or minus square root of 2 ^ 6 it is 2 ^ 3 then times this square root of 5 over 2. Then into here it will be is equal to plus or minus 2 ^ 3 it is 8 then times square root of 5 divide by 2. Here we divide by 2 and into this side divide by 2. So, here it will be is equal to 24 / 2 it is 12 plus or minus 8 / 2 it is 4 square root of 5. So, from here we have two solutions. Whereas the first value of is equal to when it is positive it is 12 plus 4 square root of 5. And the second value of here is equal to when it is negative is 12 minus 4 square root of 5. Now, from here I've got the values of Now, to get we apply this equation in terms of is equals 24 minus So, from is equal to 24 minus then here this is x1 so it will be y1 is equal to 24 minus x1 we substitute here 12 plus 4 square root of 5. So, here it will be y1 is equal to 24. Here we take negative to be minus 12 minus 4 square root of 5. So, here it will be y1 is equal to 24 minus 12 it is 12 minus 4 square root of 5. So, this is the value of x1 this is y1. Then to solve from this second solution this is x2. So, here it will be we apply is equal to this here which is 24 minus So, x2 here it will be y2 is equal to 24 minus x2 it is 12 minus 4 square root of 5. So, here it will be y2 is equal to 24. We take negative inside, so it will be minus 12. It will be plus 4 square root of 5. Then here it will be y2 is equal to 24 minus 12 it is 12. Then here plus 4 square root of 5. So, this is value of x2. This is y2. So, our conclusion x1, y1 this is from the first solution. x1 it is this, y1 it is this. So, here it is 12 plus 4 square root of 5, y1 y1 it is 12 minus 4 square root of 5. the second solution which is x2, y2 is equal to x2 it is this. Here this is y2. So, here it is 12 minus 4 square root of 5, y2 y2 it is 12 plus 4 square root of 5. So, these are all the values of and from this our problem. Now, in the next step let's check this answer if it is correct. So, to check from our problem which is plus is equal to 24 and we have times is equal to 64. So, plus it will be this plus this. So, it is 12 plus 4 square root of 5 plus it is this here from the first solution. 12 - 4 square root of 5 is it equal to 24? Now, 4 square root of 5 - 4 square root of 5 is 0. So, this and this here it is 0. Then it will be 12 + 12 is 24 is equal to 24. So, left side and right side are equal. Then we already check by using the first equation. Now, let's check by using the second equation. * so it will be this * this. So, it is 12 + 4 square root of 5 bracket * it is this here, which is 12 - 4 square root of 5. Then is it equal to this 64? Now, here to multiply 12 * 12, it is 144. Then 12 * -4 square root of 5, it will be -12 * 4 is 48 square root of 5. Then 4 square root of 5 * 12, here it will be + 4 * 12 is 48 square root of 5. Then here 4 square root of 5 * -4 square root of 5, it will be -4 square root of bracket square. Then is it equal to 64? Then here, 48 square root of 5 - 48 square root of 5 is 0. So, this and this will cancel, then it will be here 144, then 4 square it is 16. Square root of 5 square it is 5. Then is it equal to 64? Now here to be 144 minus 16 * 5 it is 80. Then is it equal to 64? Now here 4 minus 0 it is 4. 14 minus 8 it is 6. Then is it equal to This is equal 64 is equal to 64. So left side and right side are equal. Then we need to check for by using this second equation. So this first solution here it is correct. Also the second solution is correct because the value of in the first solution it is the value of in the second solution. And the value of from the first solution it is the value of in the second solution. So they interchange the values. Thank you for watching. Don't forget these steps to miss out. Subscribe to my channel and see you in the next video. Bye-bye.