anemone said:Determine the number of solutions of the simultaneous equations $x^2+y^3=29$ and $\log_3x\log_2y=1$.
Amer said:$\log_3 x = a \Rightarrow x = 3^a $
$\log_2 y = b \Rightarrow y = 2^b $
$a.b =1 $
$(3^a)^2 + (2^b)^3 = 29 $
$(3^{2/b}) + (2^{3b}) = 29 $
$f(x) = 3^{2/x} + 2^{3x} - 29 $
$f(1) = 9 + 8 - 29 < 0 $
$f(2) = 3 + 64 - 29 > 0 $ we have a zero at (1,2) interval
$f(1/2) = 3^4 + 2^{3/2} - 29 > 0 $ we have anther zero at (1/2 , 1 )
numbers larger than 2 , $2^{3x}$ is more than 29 so there is not any zero (2, infinity)
for x in (0,1/2) $3^{2/x}$ is more than 29 f(x) is positive
for negative numbers $3^{2/x} + 2^{3x} $ less than 29
so I think we have just two zeros
two solutions
The number of solutions for a system of equations can be determined by looking at the number of independent equations and unknown variables in the system. If the number of equations is equal to the number of variables, then the system has a unique solution. If there are more equations than variables, the system is overdetermined and has no solution. If there are fewer equations than variables, the system is underdetermined and has infinitely many solutions.
Yes, a system of equations can have no solution if it is overdetermined (more equations than variables) and the equations are inconsistent, meaning they contradict each other. For example, the system 2x + 3y = 5 and 2x + 3y = 7 has no solution since the two equations cannot be simultaneously satisfied.
If a system of equations is underdetermined (fewer equations than variables), it can have infinitely many solutions. This means that there is not a unique solution, but rather a range of possible solutions that satisfy the equations. For example, the system x + y = 3 and 2x + 2y = 6 has infinitely many solutions since any values of x and y that satisfy the first equation will also satisfy the second equation.
If a system of equations is determined (the same number of equations as variables), it can have exactly one solution. However, if the system is underdetermined, it can have infinitely many solutions. If the system is overdetermined, it can have no solution or exactly one solution, depending on whether the equations are consistent or inconsistent.
Determining the number of solutions for a system of equations is important because it helps us understand the behavior of the system. If a system has a unique solution, we can use that solution to find the values of the variables and solve for the unknowns. If a system has infinitely many solutions, we can use that information to find a range of possible solutions. And if a system has no solution, it tells us that the equations are inconsistent and cannot be solved. This knowledge can be useful in many fields, including physics, engineering, and economics.