M Assignment Inequalities, Linear Systems, Matrices Dr. Taylor Submit your typed up work at the assignment upload page. Support your answers with your work, indicating any tools that you used and uploading any support files. Do at least 8 of the following problems. 1. Find all x that satisfy: (a) 3x + 4 > 10 (b) 2 < 3x + 4 < 10 (c) | 3x + 4| < 10 (d) 3x + 5 < 5x + 7 2. Solve for all 3-tuples (x, y, z) simultaneously satisfying the following equations: 2x +4y +1z = 36 5x +8y +7z = 56 2x 8y +7z = 100 3. Solve for all 4-tuples (x1 , x2 , x3 , x4 ) simultaneously satisfying the following equations: 8x1 9x2 9x1 +6x2 3x1 9x2 7x2 2x3 6x3 +4x3 +8x3 5x4 +9x4 2x4 +8x4 = 100 = 60 = 52 = 135 4. We are given a system of three equations in the three unknowns x, y, and z. We transform an equation of the + cz = d into the row [a, b, c, d]. We row form ax + by 1 1 0 3 reduce and find the matrix 0 0 1 5 . Write down what the solutions are to this 0 0 0 2 system given this information or explain why there is no solution. 5. We are given a system of two equations in the three unknowns x, y, and z. We transform an equationof the form ax + by + cz = d into the row [a, b, c, d]. We row reduce and find 1 3 0 4 the matrix 0 0 1 7 . Write down what the solutions are to this system given this 0 0 0 0 information or explain why there is no solution. 6. Balance the following chemical equation: F2 + Al2 O3 AlF3 + O2 7. Balance the following chemical equation: C8 H18 + O2 CO + H2 O 8. Graph y = 3x and y = 2 in the same picture. Graph y = 3x and y = 2 in a separate picture. Explain how those pictures can be used to demonstrate that 3x < 2 has the same solution set as 3x > 2. QM Assignment Inequalities, Linear Systems, Matrices Dr. Taylor 9. Suppose we have two intersecting lines. If one has two points, one on each line, vertically aligned, what happens to the distance between the two points as the points are moved closer to the intersection of the two lines? Submit a GeoGebra file that can be used to demonstrate this. Explain how to use the file. Hint: Graph two intersecting lines, say lines a and b. Add a slider to the the picture. If it is called c, then in the input box, type x = c to define a vertical line, say line d. Use the command Intersect[a,d] and Intersect[b,d] to get two points, say E and F . The command Segment[E,F] can be helpful as well. By right-clicking on line d, one can look at its properties and hide it or make it a dashed line. 10. Submit a spreadsheet that determines whether a given proposed solution is, in fact, a solution of the following system of three equations. Explain how you set it up and how to use it. x +2y +3z = 4 4x +5y +6z = 2 7x +8y +7z = 1 What is the actual solution? You may use the rref calculator method. 11. Submit a spreadsheet that can generate all solutions (there is more than one solution) for the following system of linear equations. Explain how you set it up and how to use it. x +2y +3z = 0 4x +5y +6z = 0 7x +8y +9z = 0 Page 2