TO LINEAR EQUATIONS
Relations, Functions and Linear Equations: We know that in algebra all functions are relations, but not all relations are functions. Why? How may we determine from the graph if we have a relation, or a function?
Study the definition of
relations and functions. Use of mapping, tables, ordered sets and
graph to represent relations and functions. Learn how to apply the
vertical test to identify a function. Determine the difference
between discrete vs. continues functions. Definition of linear
equations and the Standard Form.
About Slope: The inclination of a road makes all the difference for a driver of a big truck to decide when driving downhill if he needs to apply the breaks, or slow down his truck with the motor. You might remember, or in your next trip try to see that when the road is too steep; there is a signal with a truck going downhill and a number below and a caption saying: "six degree slope or other equivalent words (or another number), slow down with the motor." and telling for how many miles this is true. They are telling the drivers of big trucks that for each one hundred horizontal feet the road raises vertically six feet. For a full loaded truck downhill this poses the challenge to apply the breaks not to accelerate due to the inertia of the big mass (that is the big weight this implies) on the truck. Problem is that the tendency of the truck is to go faster than safer to go. It needs to be slowed down. If the driver applies the breaks too often they overheat and melt the plastic plugs, gaskets, and seals and then the break liquid leaks out of the break circuit and the truck ends up without a break system to stop it. The solution is to apply a lower gear shift that slows down the truck without applying the breaks. Observe that at the end of those steep slopes you have a safety ramp at the end of a guiding red line in the road full with loose gravel to slow down to a stop any truck for those drivers that didn't learn algebra to read the sign and understanding it. You will learn about slope formula (run vs. rise and two points), falling to the
right, horizontal, vertical, negative and positive cases for the slope.
Slope of parallel and perpendicular lines.
With Systems of Linear
Equations: Your dad is deciding if getting a new cell phone contract from two competing companies. One charges a fee, and a rate per minute for each call. The other does not charge a fee, but charges a little bit more per minute. His dilemma is to find out after how many minutes the companies will charge exactly the same, knowing that before that minute one charges more than the other and after that point this switches. This problem is solved with a system of linear equations. Each situation may be represented by a straight line in the coordinate plane. Where they intersect is the point your dad wants to find. To solve this and other problems you need to learn about slope intercept form and point-slope form problems that
involve a point and slope, two points, etc. Solving systems of two
variable linear equations by substitution, linear combinations and
graphing. Introduction to special functions (step function, constant
function, identity function and absolute value function)
Two Variable Linear Inequalities and Absolute Value Inequalities. Like in the section before this one. You will get a straight line in the coordinate plane. The difference is that now this line is the boundary of two areas, where one of them won't include the line itself. One side will have that all points make true the inequality, and at the other all points will make it false. If the inequality is expressed as greater or equal, or as less or equal, then the line will be part of the solution, other wise is not part of it. You will practice solving
two variable inequalities by shading above o below the graph to
indicate the solution set.
- Systems of Linear Inequalities and Introduction to Linear Programming and Its Applications. In any game you have legal movements and actions, and you have illegal movements and actions. These second ones are called restrictions. When an engineer in a company has to decide how much rubber, how much wood, how much oil, etc... for the construction of a skateboard that needs rubber for the wheels, wood for the board, and oil to lubricate the bearings of the wheels. He needs to be careful when requesting the quantities of each, so that he doesn't end up with too much, or too many of one, and two little, or too few of other components. That is done setting up a system of inequalities in the coordinate plane. They enclose one area inside a polygon. The points of interest are the vertices of the polygon. With those coordinates they use a function that is evaluated and helps to decide the maximum or minimum of all the existent coordinates in the points of the polygon. The inequalities are the "restrictions" he needs to take into account. The function to evaluate is the optimization function to be maximized or minimized. Learn
how to solve systems of two variable linear inequalities. This
concept it is extended to introduce concepts as solution region,
feasible region, and maximum and minimum of the optimization
function. Solution of linear programming problems.