Question

Solve The Equation
4x×9y=7
4x-9y=9

Answer 1

`\large\boxed{x=(9)/(8)-(√(109))/(8),\ y=-(1)/(2)-(√(109))/(18)}\\or\\\boxed{x=(9)/(8)+(√(109))/(2),\ y=-(1)/(2)+(√(109))/(18)}`

Explanation:

`\left\{\begin{array}{ccc}4x*9y=7&(1)\\4x-9y=9&(2)\end{array}\right\\\\(2)\\4x-9y=9\qquad\text{subtract}\ 4x\ \text{from both sides}\\-9y=-4x+9\qquad\text{change the signs}\\9y=4x-9\qquad\text{substitute it to (1)}\\\\4x(4x-9)=7\qquad\text{use the distributive property}\ a(b+c)=ab+ac\\(4x)(4x)+(4x)(-9)=7\\(4x)^2-36x=7\\(4x)^2-2(4x)(4.5)=7\qquad\text{add}\ 4.5^2\ \text{to both sides}\\(4x)^2-2(4x)(4.5)+4.5^2=7+4.5^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2`

`(4x-4.5)^2=7+20.25\\(4x-4.5)=27.25\to 4x-4.5=\pm√(27.25)\\\\4x-(45)/(10)=\pm\sqrt{(2725)/(100)}\\\\4x-(45)/(10)=\pm(√(2725))/(√(100))\\\\4x-(45)/(10)=\pm(√(25\cdot109))/(10)\\\\4x-(45)/(10)=\pm(√(25)\cdot√(109))/(10)\\\\4x-(45)/(10)=\pm(5√(109))/(10)\qquad\text{add}\ (45)/(10)\ \text{to both sides}\\\\4x=(45)/(10)\pm(5√(109))/(10)`

`4x=(9)/(2)\pm(√(109))/(2)\qquad\text{divide both sides by 4}\\\\x=(9)/(8)\pm(√(109))/(8)\\\\\text{Put the values of}\ x\ \text{to (2):}\\\\9y=4\left((9)/(8)\pm(√(109))/(8)\right)-9\\\\9y=(9)/(2)\pm(√(109))/(2)-(18)/(2)\\\\9y=-(9)/(2)\pm(√(109))/(2)\qquad\text{divide both sides by 9}\\\\y=-(1)/(2)\pm(√(109))/(18)`