How Can You Solve This Modulus Equation with Square Roots and Absolute Values?

  • Thread starter Thread starter chwala
  • Start date Start date
  • Tags Tags
    Modulus
AI Thread Summary
The modulus equation 2|3x+4y-2| + 3√(25-5x+2y) = 0 requires both terms to be zero since they are non-negative. The approach taken involves isolating the square root and modulus, leading to simultaneous equations. The derived equations are 3x + 4y = 5 and -5x + 2y = -21. Solving these gives approximate values of x ≈ 3.615 and y ≈ -1.46, which satisfy the original equation. The discussion highlights the importance of ensuring both components equal zero for a valid solution.
chwala
Gold Member
Messages
2,825
Reaction score
413
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
 
Last edited:
Physics news on Phys.org
chwala said:
Solve ##2|3x+4y-2|##+##3####\sqrt{25-5x+2y}##=##0##
Both terms are non-negative so have to be zero independently, yes? The solution is the intersection of two straight lines.
 
Back
Top