We want to solve the following equations:
item (a):
[tex]x=\frac{25}{x}[/tex]To solve this one, let's start by multiplying both sides by x.
[tex]x^2=25[/tex]Now, let's take the square root of both sides
[tex]\begin{gathered} \sqrt[]{x^2}=\sqrt[]{25} \\ x=\pm5 \end{gathered}[/tex]The solutions for this equation are x = 5 and x = -5.
item (b):
[tex]x+2=\frac{6x-3}{x}[/tex]To solve this one, let's again start by multiplying both sides by x.
[tex]\begin{gathered} x(x+2)=6x-3 \\ x^2+2x=6x-3 \end{gathered}[/tex]Now, let's subtract 2x from both sides.
[tex]\begin{gathered} x^2+2x-2x=6x-3-2x \\ x^2=4x-3 \end{gathered}[/tex]Let's rewrite this equation with all terms in the right side
[tex]x^2-4x+3=0[/tex]Factorizing
[tex]\begin{gathered} x^2-4x+3=(x-1)(x-3) \\ \Rightarrow(x-1)(x-3)=0 \end{gathered}[/tex]Since this is a product of two terms, the result will be zero only if one of them is zero.
Then, we get two equations
[tex]\begin{gathered} x-1=0 \\ x-3=0 \end{gathered}[/tex]The solutions for those two equations are the solutions for our system.
[tex]\begin{gathered} x-1=0\Rightarrow x=1 \\ x-3=0\Rightarrow x=3 \end{gathered}[/tex]The solutions for this equation are x = 1 and x = 3.
item (c):
[tex]\frac{x}{x^2}=\frac{3}{x}[/tex]Let's start by solving the division in the right side
[tex]\begin{gathered} \frac{x}{x^2}=x^{1-2}=x^{-1^{}}=\frac{1}{x} \\ \Rightarrow\frac{1}{x}=\frac{3}{x} \end{gathered}[/tex]Multiplying both sides by x, we have:
[tex]1=3[/tex]Since this statement is false, this equation have no solution.
item (d):
[tex]\frac{6x^2+18x}{2x^3}=\frac{5}{x}[/tex]Let's start by multiplying both sides by x.
[tex]\frac{6x^2+18x}{2x^2}=5[/tex]Doing the division on the left side of the equality, we have:
[tex]\begin{gathered} \frac{6x^2+18x}{2x^2}=\frac{6x^2}{2x^2}+\frac{18x}{2x^2}=3+\frac{9}{x} \\ \Rightarrow3+\frac{9}{x}=5 \end{gathered}[/tex]Subtracting 3 from both sides:
[tex]\frac{9}{x}=2[/tex]Multiplying both sides by x again:
[tex]\begin{gathered} 9=2x \\ x=\frac{9}{2}=4.5 \end{gathered}[/tex]The solution for this system is x = 4.5.
