Solving Systems Of Linear Inequalities, and Introduction To Linear Programming And Its Applications

Have you wondered how is it that factories know how much material to buy when going to produce the skateboard you enjoy so much? They need to know how much wood or how many wheels? All that may be expressed with lineal equations (most of the time) the area that is enclosed in those linear equations when they are graphed contains all the possible solutions. This is linear programming. Linear programming may be a scary topic to learn if you struggle with algebra 1. How are we supposed to set up the inequalities to find the solution region? What is the feasible region? What is the meaning of optimization function? What is the use of linear programming?

Before you start the lesson, scroll down the page and review the definitions given there. There are some animated graphics as well to review forms of linear equations. While going through the lesson itself you may use the MARKER TOOLS menu to write notes or highlight parts of the solution of the problems. This is located at the upper right section of the screen. In this lesson you will find that we have stripped the topic from all the formality, while still keeping the rigor, and making use of your visual learning side. Linear programming won't be that scary experience anymore!

Lesson's Content

Lesson In PDF Format (no animations)

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Lesson's Glossary

Absolute value:  Distance of a number from zero on a number line. The distance is taken as positive all the time. For a variable: If x < 0 then –a; if x>=0 then a.

Equivalent system:  A system of equations having the same solution set as another system.

Feasible region: Given a set of inequalities with a set of constrains; the area in the graph that satisfies all the inequalities and constrains is the feasible region.

Graph of an equation in two variables:  All the points that may be graph from the solution set of the equation.

Graph of a number:   The location of a point paired with a number in the number line.

Graph of an ordered pair:  The location in the coordinate plane associated with an ordered pair of real numbers.

Inequality:  A statement formed by placing an inequality symbol between numerical or variable expressions.

Inequality symbols:  Symbols used to show the order of two real numbers.

Sides of an inequality:   The two expressions at both sides of the inequality sign in an inequality. Left and right.

Linear equation:   Any equation with all exponents = 1 regardless of the form the equation is represented.

Linear equation in two variables:   All equivalent equations to the one in the form of ax + by = c, where a, b, and c are in the set of the real numbers and a and b can't be zero at the same time. The graph is a straight line.

Linear function:  A function of the form: f(x) = mx + b.

Linear programming: The method for which we maximize or minimize a function (optimization function or objective function) by finding the solution polygon for the two variable inequalities generated by the constrains in the problem and using the coordinates in the vertices of the polygon to evaluate the optimization function. The highest value in the answer is the maximum and the lowest value is the minimum.

Objective function It is the optimization function in linear programming (the one to maximize or to minimize)

Ordered pair:   In a coordinate plane is the location of a point.

Ordinate:  The y-coordinate in an ordered pair of the coordinate plane.

Origin:   The location of the zero point in a number line.

Plot a point:   Locating and graphing an ordered pair of real numbers in the coordinate plane.

Point-slope form of an equation:  Y - Y1 = m(X-X1), where m is the slope and (X1 , Y1) the point for which the line goes through.

Slope of a line: The measure of how steep a line is. The change in y (rise) divided by the change in x (run). Slope = m.

Slope-intercept form of an equation:   The equation of a line in the form y = mx + b, where m represents the slope, and be represents the y-intercept.

Solution of an equation in two variables:  Any ordered pair of real numbers that makes the sentence true.

Solution of a system of two equations in two variables:   Ordered pair that when replaced in the equations produces a true statement for both equations.

Solve a system of two equations in x and y:   Finding all ordered pairs (x,y) which make both equations in the system true.

Standard form of a linear equation:   ax + by = c, where a, b, and c are integers and a and b are not both zero.

x-intercept:  The x-coordinate of a given point for which the graph intersects the x-axis.

y-intercept: The y-coordinate of a given point for which the graph intersects the y-axis.

Vocabulary Highlights

Slope of a line: The measure of how steep a line is. The change in y (rise) divided by the change in x (run). Slope = m

Slope of a Line

Slope-intercept form of an equation:   The equation of a line in the form y = mx + b, where m represents the slope, and be represents the y-intercept.

Slope Intercept

Point-slope form of an equation:  Y - Y1 = m(X-X1), where m is the slope and (X1 , Y1) the point for which the line goes through.

Point Slope Form

 

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