Ch 7.4 – Applications of Linear Systems Algebra 1 Ch 7.4 – Applications of Linear Systems

Objective Students will choose the best method to solve linear systems of equations

Before we begin… Thus far we have looked at 3 different ways to solve systems of linear equations… This lesson focuses on you choosing the best method given some guidelines… Even though you can solve each of the linear systems with each method…the goal here is for you to find the quickest, simplest way…

Guidelines Recall that in the Graphing Method you transformed the equations to slope-intercept form and graphed the lines. The point where the lines intersect represents the solution to the system of equations This method is useful for approximating a solution, checking the reasonableness of a solution and providing a visual model

Guidelines Recall that in the Substitution Method you solved one of the equations for one of its variables and then substituted the resulting expression into the other equation to get an equation with one variable. You then solved the equation and used the result to find the value of the other variable. This method is useful when one of the variables has a coefficient of 1 or -1.

Guidelines Recall that in the Combination Method you manipulated the equations to eliminate one of the variables. You then combined the equations, solved for the variable and used the result to determine the value of the other variable. This method is useful when none of the variables have a coefficient of 1 or -1.

Examples At this point let’s look at some examples and see if we can analyze them and determine the best method to solve based upon the guidelines…

Example #1 x + y = 300 x + 3y = 18 Take a minute and look at the guidelines and see if you can determine the best method. You should be able to explain your choice In this example you could either use the substitution method or linear combinations. It would be easy to write either variable in terms of the other or to eliminate x by multiplying either equation by -1

Example #2 3x + 5y = 25 2x – 6y =12 Again, look at the guidelines and see if you can determine the best method. You should be able to explain your choice In this example linear combinations would be the best way to solve the equations since neither variable has a coefficient of 1 or -1.

Example #3 2x + y = 0 x + y = 5 One more time…use the guidelines and see if you can determine the best method… In this example any of the three methods could be used. It would be simple to write either variable in terms of the other, both would be simple to graph, and y could be eliminated by multiplying either equation by -1

Comments The beauty of systems of equations is that you get to choose what method you are most comfortable with. The key is to analyze the equations first, then decide which method would be easiest and quickest and then follow through using the steps you learned.

Comments On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn Choose a method to solve the linear system. Explain your choice. (You do not have to solve these problems) 6x + y = 2 9x – y = 5 2x + 3y = 3 5x + 5y = 10 2x – 5y = 0 x – y = 3 3x + 2y = 10 2x + 5y = 3

Your Turn Choose a method and then solve 2x + y = 5 and x – y = 1 8x + y = 15 and 9 = 2y + 2x 100 – 9x = 5y and 0 = 5y – 9x X + 2y = 2 and x + 4y = -2 0.2x – 0.5y = -3.8 and 0.3x + 0.4y = 10.4

Your Turn Solutions Answers can vary. Make sure that you explain your reasoning! Substitution method Linear combinations Substitution or graphing method (2, 1) (4/15, 1 1/5) (1 ½ , 3) (5 5/9, 10) (6, -2) (16, 14)

Summary A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about applications of linear systems. Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you credit for doing this lesson…please see the next slide…

Credit I will add 25 points as an assignment grade for you working on this lesson… To receive the full 25 points you must do the following: Have your name, date and period as well a lesson number as a heading. Do each of the your turn problems showing all work Have a 1 paragraph summary of the lesson in your own words Please be advised – I will not give any credit for work submitted: Without a complete heading Without showing work for the your turn problems Without a summary in your own words…