3.4. If \( y=x^{3}+1 \), calculate \( y \) when \( x=4 \) 3.5. Thandi is 6 years older than Sophie. In three years' time Thandi will be twice as old as Sophie. How old is Thandi now?

3.4. Given the equation \( y=x^{3}+1 \), we need to calculate the value of \( y \) when \( x=4 \). Substitute \( x=4 \) into the equation \( y=x^{3}+1 \): \[ y=4^{3}+1 \] Calculate the value of \( y \): \[ y=64+1=65 \] Therefore, when \( x=4 \), \( y=65 \). 3.5. Let's denote Sophie's current age as \( S \) and Thandi's current age as \( T \). From the information given: 1. Thandi is 6 years older than Sophie: \( T = S + 6 \) 2. In three years' time, Thandi will be twice as old as Sophie: \( T + 3 = 2(S + 3) \) We can solve these two equations to find the current age of Thandi. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}T=S+6\\T+3=2\left(S+3\right)\end{array}\right.\) - step1: Substitute the value of \(T:\) \(S+6+3=2\left(S+3\right)\) - step2: Simplify: \(S+9=2\left(S+3\right)\) - step3: Expand the expression: \(S+9=2S+6\) - step4: Move the variable to the left side: \(S+9-2S=6\) - step5: Subtract the terms: \(-S+9=6\) - step6: Move the constant to the right side: \(-S=6-9\) - step7: Subtract the numbers: \(-S=-3\) - step8: Change the signs: \(S=3\) - step9: Substitute the value of \(S:\) \(T=3+6\) - step10: Simplify: \(T=9\) - step11: Calculate: \(\left\{ \begin{array}{l}S=3\\T=9\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}S=3\\T=9\end{array}\right.\) - step13: Rewrite: \(\left(S,T\right) = \left(3,9\right)\) The solution to the system of equations is \( S = 3 \) and \( T = 9 \). Therefore, Sophie is currently 3 years old, and Thandi is currently 9 years old.