SIMULTANEOUS EQUATIONS INVESTIGATION In this task, you will investigate the patterns in the intersection of two lines 𝑦 = 𝑥 and 𝑦 = −𝑥 + 2 Then you will be asked to prove your conjectures and to broaden the scope of the investigation to include other lines. Method 1. Consider the lines 𝑦 = 𝑥 𝑎𝑛𝑑 𝑦 = −𝑥 + 2  Use method by elimination to solve the simultaneous equations. You must show all working.  Using technology, find the intersections of the lines given by equations and graph the pairs of lines on Cartesian Plane. 2. Solve the following pairs of equations 𝑦 = 𝑥 + 𝑘 𝑎𝑛𝑑 𝑦 = −𝑥 + 2 + 𝑘 using the substitution and graphical methods. Check the solutions. Consider various k ≥ 0 values . Make an initial conjecture about the solutions of these simultaneous equations. 3. Investigate your conjecture for any real value of k. Refine your initial conjecture as necessary. 4. Does your conjecture hold if the intersecting lines are changed? Modify your conjecture, if necessary and prove it.