Q1. a) Solve i. \( x^{2}+y^{A^{2}}=4 \) and \( y=2 \) ii. \( x^{2}+y^{7}=4 \) and \( y=x+1 \) b) what does your answer in (a) represent?
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The Deep Dive
To solve the equations, let's start with (i). Given \( y=2 \), substituting it into the first equation gives us \( x^{2}+2^{A^{2}}=4 \). This simplifies to \( x^{2}=4-2^{A^{2}} \). The solutions for \( x \) depend on the value of \( A \). For (ii), substituting \( y=x+1 \) into the second equation gives us \( x^{2}+(x+1)^{7}=4 \). This is a complex polynomial equation that may need numerical methods or graphical solutions to find values of \( x \). The answers in (a) represent points of intersection between curves defined by the equations. In geometry, these points often reveal solutions to systems of equations, showcasing where one equation's output fits into the context of another.
