\( \quad=15 x a \) \( x^{10}=10 x^{9} \) \( x^{3}+x^{2}-100=1 \) using of Itration method \( x_{0}=0.5 \)

To solve the given system of equations using the iteration method, we will start with the initial guess \( x_{0} = 0.5 \) and iteratively update the value of \( x \) until the solution converges. Given equations: 1. \( x^{10} = 10x^{9} \) 2. \( x^{3} + x^{2} - 100 = 1 \) Let's start with the first equation: 1. \( x^{10} = 10x^{9} \) We can rewrite this equation as: \( x^{10} - 10x^{9} = 0 \) Now, let's solve this equation using the iteration method. Solve the equation by following steps: - step0: Solve for \(x\): \(x^{10}-10x^{9}=0\) - step1: Factor the expression: \(x^{9}\left(x-10\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x^{9}=0\\&x-10=0\end{align}\) - step3: Set the base equal to 0: \(\begin{align}&x=0\\&x=10\end{align}\) - step4: Rewrite: \(x_{1}=0,x_{2}=10\) The solution to the equation \( x^{10} - 10x^{9} = 0 \) is \( x = 0 \) or \( x = 10 \). Now, let's move to the second equation: 2. \( x^{3} + x^{2} - 100 = 1 \) We can rewrite this equation as: \( x^{3} + x^{2} - 101 = 0 \) Now, let's solve this equation using the iteration method. Solve the equation(The complex numbers system) by following steps: - step0: Solve for \(x\) in the complex numbers system: \(x^{3}+x^{2}-101=0\) - step1: Calculate: \(\begin{align}&x\approx 4.346402\\&x\approx -2.673201+4.011435i\\&x\approx -2.673201-4.011435i\end{align}\) - step2: Rewrite: \(x_{1}\approx -2.673201-4.011435i,x_{2}\approx -2.673201+4.011435i,x_{3}\approx 4.346402\) - step3: Remove the complex number(s): \(x\approx 4.346402\) The solution to the equation \( x^{3} + x^{2} - 101 = 0 \) is approximately \( x \approx 4.346402 \). Therefore, the solution to the system of equations using the iteration method is \( x \approx 4.346402 \).