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x×y=3x \times y = 3 and x+y=5x + y = 5 Find xx and yy

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Q. x×y=3x \times y = 3 and x+y=5x + y = 5 Find xx and yy
  1. Given Equations: We are given two equations:\newline11) x×y=3x\times y=3 (Equation 11)\newline22) x+y=5x+y=5 (Equation 22)\newlineWe will solve these equations simultaneously to find the values of xx and yy.
  2. Express yy in terms: From Equation 22, we can express yy in terms of xx:y=5−xy = 5 - x (Equation 33) Now we have yy as a function of xx, which we can substitute into Equation 11.
  3. Substitute yy into Equation: Substitute yy from Equation 33 into Equation 11:x×(5−x)=3x\times(5 - x) = 3Expand the left side of the equation:5x−x2=35x - x^2 = 3
  4. Rearrange quadratic equation: Rearrange the equation into a standard quadratic form: x2−5x+3=0x^2 - 5x + 3 = 0 This is a quadratic equation in terms of xx.
  5. Solve quadratic equation: Solve the quadratic equation for xx. We can use the quadratic formula, x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=1a = 1, b=−5b = -5, and c=3c = 3.\newlinex=5±25−4×1×32×1x = \frac{5 \pm \sqrt{25 - 4\times1\times3}}{2\times1}\newlinex=5±25−122x = \frac{5 \pm \sqrt{25 - 12}}{2}\newlinex=5±132x = \frac{5 \pm \sqrt{13}}{2}
  6. Find corresponding y values: We have two possible solutions for xx:x=5+132x = \frac{5 + \sqrt{13}}{2} or x=5−132x = \frac{5 - \sqrt{13}}{2}Now we will find the corresponding y values for each xx using Equation 33.
  7. Check first pair of solutions: For x=5+132x = \frac{5 + \sqrt{13}}{2}, find yy using Equation 33:\newliney=5−(5+132)y = 5 - \left(\frac{5 + \sqrt{13}}{2}\right)\newliney=10−5−132y = \frac{10 - 5 - \sqrt{13}}{2}\newliney=5−132y = \frac{5 - \sqrt{13}}{2}
  8. Check second pair of solutions: For x=5−132x = \frac{5 - \sqrt{13}}{2}, find yy using Equation 33:\newliney=5−(5−132)y = 5 - \left(\frac{5 - \sqrt{13}}{2}\right)\newliney=10−5+132y = \frac{10 - 5 + \sqrt{13}}{2}\newliney=5+132y = \frac{5 + \sqrt{13}}{2}
  9. Check second pair of solutions: For x=5−132x = \frac{5 - \sqrt{13}}{2}, find yy using Equation 33:\newliney=5−(5−132)y = 5 - \left(\frac{5 - \sqrt{13}}{2}\right)\newliney=10−5+132y = \frac{10 - 5 + \sqrt{13}}{2}\newliney=5+132y = \frac{5 + \sqrt{13}}{2}We have found two pairs of solutions for xx and yy:\newline11) x=5+132x = \frac{5 + \sqrt{13}}{2} and y=5−132y = \frac{5 - \sqrt{13}}{2}\newline22) x=5−132x = \frac{5 - \sqrt{13}}{2} and y=5+132y = \frac{5 + \sqrt{13}}{2}\newlineWe need to check if these solutions satisfy both original equations.
  10. Check second pair of solutions: For x=5−132x = \frac{5 - \sqrt{13}}{2}, find yy using Equation 33:\newliney=5−(5−132)y = 5 - \left(\frac{5 - \sqrt{13}}{2}\right)\newliney=10−5+132y = \frac{10 - 5 + \sqrt{13}}{2}\newliney=5+132y = \frac{5 + \sqrt{13}}{2}We have found two pairs of solutions for xx and yy:\newline11) x=5+132x = \frac{5 + \sqrt{13}}{2} and y=5−132y = \frac{5 - \sqrt{13}}{2}\newline22) x=5−132x = \frac{5 - \sqrt{13}}{2} and y=5+132y = \frac{5 + \sqrt{13}}{2}\newlineWe need to check if these solutions satisfy both original equations.Check the first pair of solutions in both equations:\newlineFor x=5+132x = \frac{5 + \sqrt{13}}{2} and y=5−132y = \frac{5 - \sqrt{13}}{2}:\newlineEquation 11: yy33 (True)\newlineEquation 22: yy44 (True)
  11. Check second pair of solutions: For x=5−132x = \frac{5 - \sqrt{13}}{2}, find yy using Equation 33:\newliney=5−(5−132)y = 5 - \left(\frac{5 - \sqrt{13}}{2}\right)\newliney=10−5+132y = \frac{10 - 5 + \sqrt{13}}{2}\newliney=5+132y = \frac{5 + \sqrt{13}}{2}We have found two pairs of solutions for xx and yy:\newline11) x=5+132x = \frac{5 + \sqrt{13}}{2} and y=5−132y = \frac{5 - \sqrt{13}}{2}\newline22) x=5−132x = \frac{5 - \sqrt{13}}{2} and y=5+132y = \frac{5 + \sqrt{13}}{2}\newlineWe need to check if these solutions satisfy both original equations.Check the first pair of solutions in both equations:\newlineFor x=5+132x = \frac{5 + \sqrt{13}}{2} and y=5−132y = \frac{5 - \sqrt{13}}{2}:\newlineEquation 11: yy33 (True)\newlineEquation 22: yy44 (True)Check the second pair of solutions in both equations:\newlineFor x=5−132x = \frac{5 - \sqrt{13}}{2} and y=5+132y = \frac{5 + \sqrt{13}}{2}:\newlineEquation 11: yy77 (True)\newlineEquation 22: yy88 (True)

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