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To solve a system means to find the ordered pairs that satisfy both inequalities. Graph and find the intersection of the two inequalities. If they do not intersect then there is no solution to the system. A system of more than two linear inequalities can have a solution that is a bounded set of points. A bounded set of all points on or inside a convex polygon graphed on a coordinate plane is said to be a polygonal convex set.

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Ex 1: Solve the system. x 0 y 0 x + y 5 Name the coordinates of the polygonal convex set.

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Ex 2 Bob is a graphic designer who makes greeting cards. His startup cost will be $1500 plus $0.40 per card. IN order for him to remain competitive with large companies, he must sell his cards for no more than $1.70 each. How many cards must he sell in order to make a profit?

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An expression whose value depends on two variables is a function of two variables. 6x +7y – 9 is a function of x and y and can be written f(x,y) = 6x+7y – 9. The expression f(3,5) would stand for the value of the function f when x is 3 and y is 5.

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Sometimes it is necessary to find the max or min value that a function has. Since its impossible to evaluate a function for the infinitely many points in the polygonal convex set- we have the Vertex Theorem. Vertex theorem – the maximum or minimum value of f(x, y) = ax + by + c on a polygonal convex set occurs at a vertex of the polygonal boundary.

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Ex 3 Find the maximum and minimum values of f(x) = y – 2x + 5 for the polygonal convex set determined by the system of inequalities. x 1 y 2 y 8 x + y 5 2x + y 14

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Key Concepts for Sect. 7.1 *A system of equations is two or more equations in two or more variables. *Numerically, a solution to a system of equations.

Key Concepts for Sect. 7.1 *A system of equations is two or more equations in two or more variables. *Numerically, a solution to a system of equations.

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