5.5. Exercises

Solve the following systems of linear equations (with variables $x,y,z,$ and $w$) using the augmented matrix method. No other method will be given credit.

1. (i)

2. (ii)

3. (iii)

4. (iv)

5. (v)

6. (vi)

7. (vii)

8. (viii)

9. (ix)

10. (x)

11. (xi)

For each system of equations in Exercise 5.5.1 using at most 3 variables interpret your answer geometrically.

At the end of the Winter Olympics, Canada, Russia, and Great Britain tally their gold, silver, and bronze medals. Canada has twice as many gold as it has silver, and equal numbers of silver and bronze medals. Russia, has twice as many silver as gold, and an equal number of gold and bronze medals. Team GB won an equal number of silver and bronze, and (for the first time ever) two gold medals. Combining all three countries together, there are 18 gold, 19 silver, and 13 bronze.

1. (i)

   Convert this question into a system of three equations with three variables.

2. (ii)

   Convert that system to an augmented matrix, and row reduce it.

3. (iii)

   How many of each type of medal did each of these countries win?

The shop Regular-Convex-Polyhedra-R-Us has in its collection several shapes: tetrahedra, cubes, dodecahedra, and icosohedra. When taking inventory, an employee calculated they had a combined total of 100 vertices, 174 edges, and 94 faces. Recall that each cube has 8 vertices, 12 edges, and 6 faces (ask Google for the values of the other shapes… they have been known to humans for several thousand years). The employee also noticed there were the same number of cubes as tetrahedra.

1. (i)

   Set up a corresponding system of equations.

2. (ii)

   Convert that system into an augmented matrix, and row reduce it.

3. (iii)

   How many of each shape were in the collection?

Define matrices as follows:

For which values of $\lambda$ is it possible to find real numbers $x,y,$ and $z$, not all of which are zero, such that the following matrix equation is satisfied

Make up a situation in which one might need to solve the following system of linear equations:

Be creative!