Solving Linear Programming Word Problems

Solving Linear Programming Word Problems-7
Here are some examples: The easiest way to graph this is with the “cover up” or intercepts method, since we have variables on one side and the constant on the other.To graph using the intercepts method, cover up the \(4y\) (making \(y=0\)) and we see the \(x\) intercept is , since \(-4\times -3=12\).

Here are some examples: The easiest way to graph this is with the “cover up” or intercepts method, since we have variables on one side and the constant on the other.

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Here are what some typical Systems of Linear Inequalities might look like in Linear Programming: We’ll graph the equations as equalities, and shade “up” or shade “down”.

The easiest way to graph the first two inequalities is with the intercepts or “cover up” method.

Finally, we substitute these ordered pairs into our objective equations and select the maximum or minimum value, based on the desired result.

Additionally, we will utilize all of our skills of solving system of equations, such as the graphing method, substitution method, and the elimination method to aid us in solving linear programming word problems.

Note that you can always plug in an \((x,y)\) ordered pair to see if it shows up in the shaded areas (which means it’s a solution), or the unshaded areas (which means it’s not a solution.) For an example of this, see the first inequality below.

With “\(\)” inequalities, we draw a dashed (or dotted) line to indicate that we’re not including that line (but everything up to it), whereas with “\(\le\)” and “\(\ge\)”, we draw a regular line, to indicate that we are including it in the solution.So this means we need to find a way of using these limitations to our advantage, like finding the optimum amount of money, space, time, etc., to accomplish our goals. First, we must identify all constraints, by creating a system of inequalities.Then we must identify the Objective Function, which is the equation we want to maximize or minimize.Before we start Linear Programming, let’s review Graphing Linear Inequalities with Two Variables.Let’s go back and revisit graphing linear inequalities on the coordinate system.To remember this, I think about the fact that “\(\)” have less pencil marks than “\(\le\)” and “\(\ge\)”, so there is less pencil used when you draw the lines on the graph.You can also remember this by thinking the line under the “\(\le\)” and “\(\ge\)” means you draw a solid line on the graph.In real life, we are subject to constraints or conditions.We only have so much money for expenses; there is only so much space available; there is only so much time. Nothing more than taking several linear inequalities that all relate to some situation, and finding the “best” value under the given conditions.Next, we will graph the system of inequalities and find the feasible solution, which is the shaded or overlapping region common to all conditions.Then we will locate all vertices and corners of this feasible solution, as Purple Math accurately states.


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