Solve the equation, where the left-hand side of equation is denoted as determinant

1. Solve the equation, where the left-hand side of equation is denoted as determinant.

Solution.

Transform the determinant on the basis of properties of determinant into one that contains a row or a column with all entries 0 except one.

Now, expand the determinant by the entries of the second row

The equation can be written in the simpler form:

, , whence,

Thus, this equation has four roots:

2. A travel agency will plan tours for groups of size 25 or larger. If the group contains exactly 25 people, the cost is $300 per person. However, each person’s cost is reduced by $10 for each additional person above the 25. What size group will produce the largest revenue for the agency?

Solution:

The total revenue is

R=(number of people)(cost per person).

Let x represents additional people, the number of people will be 25 + x and the cost per person will be dollars.

Then the total revenue will be a function of x

or

This function will have its maximum where and the solution to is . Thus adding 2.5 people to the group should maximize the total revenue. But we cannot add half a person. So we will test the total revenue function for 27 people and 28 people. This will determine the most profitable number because R(x) is concave downward for all x.

For x=2 (giving 27 people) we get For x=3 (giving 28 people) we get Note that both 27 or 28 people give the same total revenue and this revenue is greater than the revenue for 25 people. Thus the revenue is maximized at either 27 or 28 people in the group.

3. Given the equations of the altitudes of a triangle and and given the coordinates of the vertice of a triangle . Find the equations of the sides of the triangle.

Solution.

Point does not lie on the altitudes because its coordinates does not satisfy the equations of the altitudes:

This implies that altitudes, which are given, passes through two other vertices and (see Figure).

Let denote its and .

.

Let altitude has the

equation and

altitude has the

equation .

Since , the

equation may be found

as the equation of the line

that is perpendicular to and passes through the point .

— the equation of the side

Similarly, the equation may be found as the equation of the line that is perpendicular to and passes through the point.

— the equation of the side Next step is to find the coordinates of the point and C as the points of intersection of the lines AB, BC and AC,DC, correspondingly.

B:

C:

Finally, let’s find the equation of the sides BC as the equation of the line that passes through the points B and C.

,

Answer:

— the equation of the side