1) The system 3x-y=9 and x+2y=2 may be solved algebraically using several different methods. Show the first step in solving the system using each method below.
a)Substitution of y
b)Elimination of x.
2) Tell how many solutions there are for a system of linear equations whose graph is described below.
a. A pair of parallel line
b. A single line
c. A pair of lines that are not parallel
3) A student proposed the following method of sketching the solution to a system of two inequalities. “First, I shade the region of points that satisfies the first inequality. Then, I shade the region of points that satisfy the second inequality. The solution consists of all points that have been shaded.” Do you agree? Explain. Support your answer using the system 5x-3y greater than 1 and 3x+4y less than or equal to 18.
4) Preparation and packaging takes 0.2 hours for each box of 12-inch pizzas and 0.25 hours for each box of 16-inch pizzas. You pay the staff no more than 240 hours of labor each week.
The staff must meet the company produce at least 1,000 boxes of pizza per week.
It will make a profit of $2 for each box of 12-inch pizzas and $4 for each box of 16-inch pizzas. How many pizzas of each size must be produced in order to maximize the profit.
Let x represent the number of 12-inch pizzas made and let y represent the number of 16-inch pizzas made.
- Write an inequality for each constraint. (Hint: There are 4.)
- Graph the feasible region that represents the constraints in (a). (Note, you must submit a graph to earn full credit.)
- Locate and label the vertices for the feasible region.
- Write the objective function.
Find the number of each type of pizza made that will maximize the profit.
1) 3x-y=9 ----------------(i)
a) From equation (i) we get:
Substitute the value of y in equation (ii):
Now, simplify and solve for x:
b) To eliminate x multiply equation (ii) by 3 and subtract from equation (i):
Therefore, by both substitution and elimination method the solutions are x=20/7, y=-3/7.
2) Solutions to systems of equations are points which all equations have in common.
a. Since parallel lines never cross, then there can be no intersection; that is, for a system of equations that graphs as parallel lines, there can be no solution. This is called an "inconsistent" system of equations, and it has no solution
b. A single line has only one equation and would thus have infinitely many solutions. Because it would have all it's points in common with itself.This is called a "dependent" system, and the "solution" is the whole line.
c. A pair of non-parallel lines would have only one solution because two non-parallel lines can only intersect at a single point. This is called an "independent" system of equations, and the solution is always some x,y-point.