- #1
chwala
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- Homework Statement
- ##2|3x+4y-2|##+##3####\sqrt{25-5x+2y}##=##0##
- Relevant Equations
- Modulus
I am trying to go through my old notes ...i came across this question,i do not have the solution.
Solve ##2|3x+4y-2|##+##3####\sqrt{25-5x+2y}##=##0##
Ok my approach on this;
##3####\sqrt{25-5x+2y}##=##-2|3x+4y-2|##
##9(25-5x+2y)=4(3x+4y-2)^2##
##9(25-5x+2y)=4(9x^2+16y^2+24xy-12x-16y+4)##
##9(25-5x+2y)=4[(3x+4y)(3x+4y)-4(3x+4)+4]##
Let ##p=(3x+4y)## and ##m=(2y-5x)##
Then it follows that
##9(25+m)=4(p^2-4p+4)##
##\frac {9}{4}##=##\frac{p^2-4p+4}{25+m}##
ok at this part i equated the numerators i.e
##(p-2)^2=9##
##→p-2=±3##, ##p=5## or ##p=-1##
and on equating the denominator, i have ##4=25+m##→##m=-21##
From this using ##p=5##, and substituting in the problem, i end up with the simultaneous equation;
##3x+4y=5##
##-5x+2y=-21##,
##x≈3.615## and ##y≈-1.46##, i substituted these values into the original equation and they both satisfy the equation...
Any comments or better way of approaching the problem? Cheers guys
Solve ##2|3x+4y-2|##+##3####\sqrt{25-5x+2y}##=##0##
Ok my approach on this;
##3####\sqrt{25-5x+2y}##=##-2|3x+4y-2|##
##9(25-5x+2y)=4(3x+4y-2)^2##
##9(25-5x+2y)=4(9x^2+16y^2+24xy-12x-16y+4)##
##9(25-5x+2y)=4[(3x+4y)(3x+4y)-4(3x+4)+4]##
Let ##p=(3x+4y)## and ##m=(2y-5x)##
Then it follows that
##9(25+m)=4(p^2-4p+4)##
##\frac {9}{4}##=##\frac{p^2-4p+4}{25+m}##
ok at this part i equated the numerators i.e
##(p-2)^2=9##
##→p-2=±3##, ##p=5## or ##p=-1##
and on equating the denominator, i have ##4=25+m##→##m=-21##
From this using ##p=5##, and substituting in the problem, i end up with the simultaneous equation;
##3x+4y=5##
##-5x+2y=-21##,
##x≈3.615## and ##y≈-1.46##, i substituted these values into the original equation and they both satisfy the equation...
Any comments or better way of approaching the problem? Cheers guys
Last edited: