Week 1, Day 1

Today I learned how to solve for variables through substitution. The first step is to isolate the variable. Once you have determined its value, subsitute it for the same variable in the second equation. By basic computation, you should be able to identify the value of each variable. If the solution is, for example, 2 = 8, then there is no solution because the statement is mathematically false. If the solution is, for example, 3 = 3, then all solutions are real numbers. Factoring may also come into play if, for instance, you get the equation x^2 - 2 = 0, in which case you would remove the x and get x (x - 2) = 0. Setting each quantity equal to zero, you would conclude that there are two solutions, 0 and 2. The following are example problems demonstrating how to solve for variables through substitution: 

Example 1: 

x + y = 4, x - y = -2

1) x = 4 - y 

2) (4 - y) - y = -2

4 - 2y = -2

-2y = -6

y = 3

3) x + 3 = 4

x = 1

Example 2:

x^2 - x - y = 1

-x + y = -1

1) y = -1 + x

2) x^2 - x - (-1 + x) = 1

x^2 - x + 1 - x = 1

x^2 - 2x + 1 = 0 

x^2 - 2x = 0 

x (x - 2) = 0 

x = 0, 2

3) If x = 0, then 

-0 + y = -1

y = -1

If x = 2, then

2 + y = -1

y = -3

y = -1, -3

I also learned two equations dealing with the concept of "breaking-even."

1) Total cost = cost per unit x # of units sold + initial cost

2) Total revenue = price per unit x # of units sold

3) Break-even if total cost = total revenue