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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?

Ask by Osborne Welch. in Papua New Guinea
Mar 12,2025

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**Part (a):** **i.** Given the equations: \[ x^{2} + y^{A^{2}} = 4 \] \[ y = 2 \] Substituting \( y = 2 \) into the first equation: \[ x^{2} + (2)^{A^{2}} = 4 \] \[ x^{2} = 4 - 2^{A^{2}} \] \[ x = \pm \sqrt{4 - 2^{A^{2}}} \] **ii.** Given the equations: \[ x^{2} + y^{7} = 4 \] \[ y = x + 1 \] Substituting \( y = x + 1 \) into the first equation: \[ x^{2} + (x + 1)^{7} = 4 \] \[ x^{2} + x^{7} + 7x^{6} + 21x^{5} + 35x^{4} + 35x^{3} + 21x^{2} + 7x + 1 = 4 \] \[ x^{7} + 7x^{6} + 21x^{5} + 35x^{4} + 35x^{3} + 22x^{2} + 7x - 3 = 0 \] This is a 7th-degree polynomial equation, which is complex to solve analytically. Numerical methods or graphing techniques would be required to approximate the solutions. **Part (b):** The solutions to the system of equations in part (a) represent the points of intersection between the two curves defined by the equations. Specifically: - In part (a)i, the solutions \( x = \sqrt{4 - 2^{A^{2}}} \) and \( x = -\sqrt{4 - 2^{A^{2}}} \) indicate the points where the hyperbola-like curve \( x^{2} + y^{A^{2}} = 4 \) intersects the horizontal line \( y = 2 \). - In part (a)ii, the solutions to the 7th-degree polynomial equation represent the points where the 7th-degree curve \( x^{2} + y^{7} = 4 \) intersects the straight line \( y = x + 1 \). These intersection points are significant in various applications, such as determining equilibrium points in physics, solving optimization problems in economics, or analyzing geometric intersections in engineering.

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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.

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