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Question: Solve the following equations: \( 4x - 3y = 1, \\\ 12xy + 13{y^2} = 25. \\\ \)...

Solve the following equations:
4x−3y=1, 12xy+13y2=25.  4x - 3y = 1, \\\ 12xy + 13{y^2} = 25. \\\

Explanation

Solution

Hint: - Substitute the value from 1st{1^{st}}equation into2nd{2^{nd}}equation.

Given equations is
4x−3y=1.............................(1) 12xy+13y2=25...........................(2)  4x - 3y = 1.............................\left( 1 \right) \\\ 12xy + 13{y^2} = 25...........................\left( 2 \right) \\\
From equation 1
y=4x−13...................(3)y = \dfrac{{4x - 1}}{3}...................\left( 3 \right)
Put this value ofyyin equation 2
12x(4x−13)+13(4x−13)2=25 4x(4x−1)+139(4x−1)2=25  12x\left( {\dfrac{{4x - 1}}{3}} \right) + 13{\left( {\dfrac{{4x - 1}}{3}} \right)^2} = 25 \\\ 4x\left( {4x - 1} \right) + \dfrac{{13}}{9}{\left( {4x - 1} \right)^2} = 25 \\\
Multiply by 9 in equation
36x(4x−1)+13(4x−1)2=225 144x2−36x+13(16x2+1−8x)=225 352x2−140x−212=0  36x\left( {4x - 1} \right) + 13{\left( {4x - 1} \right)^2} = 225 \\\ 144{x^2} - 36x + 13\left( {16{x^2} + 1 - 8x} \right) = 225 \\\ 352{x^2} - 140x - 212 = 0 \\\
Divide by 4 in the equation
88x2−35x−53=088{x^2} - 35x - 53 = 0
Divide the equation by 88.
x2−3588x−5388=0 x2−x+5388x−5388=0  {x^2} - \dfrac{{35}}{{88}}x - \dfrac{{53}}{{88}} = 0 \\\ {x^2} - x + \dfrac{{53}}{{88}}x - \dfrac{{53}}{{88}} = 0 \\\
So, factorize this equation

(x−1)(x+5388)=0 ⇒x−1=0⇒x=1 ⇒x+5388⇒x=−5388  \left( {x - 1} \right)\left( {x + \dfrac{{53}}{{88}}} \right) = 0 \\\ \Rightarrow x - 1 = 0 \Rightarrow x = 1 \\\ \Rightarrow x + \dfrac{{53}}{{88}} \Rightarrow x = - \dfrac{{53}}{{88}} \\\
Now, from equation 3
y=4x−13y = \dfrac{{4x - 1}}{3}
When
x=1 ⇒y=4−13=33=1  x = 1 \\\ \Rightarrow y = \dfrac{{4 - 1}}{3} = \dfrac{3}{3} = 1 \\\
When
x=−5388 y=4(−5388)−13=−5322−13=−7522×3=−2522  x = - \dfrac{{53}}{{88}} \\\ y = \dfrac{{4\left( { - \dfrac{{53}}{{88}}} \right) - 1}}{3} = \dfrac{{ - \dfrac{{53}}{{22}} - 1}}{3} = \dfrac{{ - 75}}{{22 \times 3}} = - \dfrac{{25}}{{22}} \\\
So, the required solution for the given equation is (1,1), (−5388,−2522)\left( {1,1} \right),{\text{ }}\left( { - \dfrac{{53}}{{88}}, - \dfrac{{25}}{{22}}} \right)

Note: - whenever we face such types of question always put the value ofxxoryy from simple equation into complex equation, then simplify the equation and find out the value of xxoryy, then put these values in the first equation we will get the required solution of the equations.