Systems of 2 Equations
How you write your answer for a system of 2 equations
(x, y)
How you write your answer for a system of 3 equations
(x, y, z)
This is how you determine the solution for a system of inequalities
The double shaded region
The first step in solving a system word problem
Define your variables
After defining variables, this is the first equation your should write when solving a linear programming problem
Objective Function
Solve for x
7x - 7y = -7
y = 2x
x = 1
Solve for x:
2x + y + 3z = 10
5y + z = -10
7z = 35
x=-1
When do you use a dashed line instead of a solid line?
> or <
Write an inequality to express the following:
Sherry has $100 to spend on pencils and paper. Pencils cost $2 per pack and paper costs $5 per pack.
2x + 5y <= 100
What is the feasible region?
Where the shading for all constraints overlap
If solving a system of 2 equations graphically, where do you find the solution?
The intersection of the two lines.
Solve the system of equations:
y = -x - 5
-4x - 2z = 6
x = 3y + 11
(-1, -4, -1)
How would you shade the linear inequality?
y < -2x +4
Shade below/left
You are running a concession stand at a basketball game. You are selling hot dogs & sodas. Each hot dog costs $1.50 and each soda costs $0.50. At the end of the night you made a total of $78.50. You sold a total of 87 hot dogs and sodas combined, 52 of which are sodas. How many hot dogs did you sell?
35 hot dogs
x=hot dogs, y=sodas
1.5x + 0.5y = 78.50
x + y = 87
y = 52
How do you find your corner points?
Intersection of lines for your feasible region
Solve the system of equations
-8x + 2y = 0
16x - 4y = 4
No Solution
Solve the system of equations:
x + 4y - 5z = 15
-4x - 2y + 5z = -18
3x - 5z = 9
(3, 3, 0)
When is there no solution to a system of inequalities? Show graphically or explain.
Parallel lines shaded opposite directions
Ruby and her children went into a movie theater and she bought $42 worth of bags of popcorn and drinks. Each bag of popcorn costs $6 and each drink costs $4. She bought a total of 8 bags of popcorn and drinks altogether. Determine the number of bags of popcorn and the number of drinks that Ruby bought.
3 Drinks
5 Bags of Popcorn
If every shirt (x) sold produces $5 in profit and every hoodie (y) sold produces $12 in profit, write an objective function.
5x + 12y
Solve the system of equations
y = 6x - 20
18x - 3y = 60
Infinite Solutions
Solve the system of equations for z:
3x - 5y - 3z = 1
-x + 5y - 3z = -9
2x - 5y - 2z = -1
z = 3
(5, 1, 3)
Is (6, 5) a solution to the system of inequalities?
y < x + 1
y < 1.5x - 4
No
5 < 5 is false (5=5)
Caleb went to the grocery store and purchased cans of soup and frozen dinners. Each can of soup has 200 mg of sodium and each frozen dinner has 500 mg of sodium. Caleb purchased a total of 15 cans of soup and frozen dinners which collectively contain 4500 mg of sodium. Determine the number of cans of soup purchased and the number of frozen dinners purchased.
10 cans of soup & 5 frozen dinners
What two things should be stated in the conclusion?
Max value & how to achieve it.