CHAPTER 5 LINEAR SYSTEMS

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1 CHAPTER 5 LINEAR SYSTEMS Systems of Linear equations have either one solution (independent), no solutions (inconsistent), or infinitely many solutions (dependent). An independent system is the case when the equations represent two intersecting lines. The solution is the intersection point. An inconsistent system is the case when the equations represent two parallel lines. They never intersect. A dependent system is the case when the equations represent the same line. Therefore they intersect everywhere on the line. Independent system: one solution and one intersection point Inconsistent system: no solution and no intersection point Dependent system: the solution is the whole line y = x + 3 y = - ½ x y = x+ 3 y = x - 2 y = x + 3 2y - 2x= 6

2 SYSTEMS OF LINEAR EQUATIONS A system of linear equations is a set of two equations of lines. A solution of a system of linear equations is the set of ordered pairs that makes each equation true. That is, the set of ordered pairs where the two lines intersect. If the system is INDEPENDENT, there is ONE SOLUTION, an ordered pair (x,y) If the system is DEPENDENT, there is INFINITELY MANY SOLUTIONS, all (x,y) s that make either equation true (since both equations are essentially the same in this case. If the system is INCONSISTENT, there are NO SOLUTIONS, because the two equations represent parallel lines, which never intersect. GRAPHING METHOD. Graph each line. This is easily done be putting them in slope-intercept form, y = mx + b. The solution is the point where the two lines intersect. SUBSTITUTION METHOD. 2x y = 5 3x + y = 5 Choose equation to isolate a variable to solve for. In this system, solving for y in the second equation makes the most sense, since y is already positive and has a coefficient of 1. This second equation turns into y = -3x + 5 Now that you have an equation for y in terms of x, substitute that equation for y in the first equation in your system. Substitute y = -3x + 5 in 2x y =5 2x (-3x + 5) = 5 Simplify and solve for x. 2x + 3x 5 = 5 5x 5 = 5 5x = 10 x = 2 ADDITION METHOD 5x + 2y = -9 12x 7y = 2 Eliminate one variable by finding the LCM of the coefficients, then multiply both sides of the equations by whatever it takes to get the LCM in one equation and LCM in the other equation. After this we can add both equations together and eliminate a variable. Let s choose to eliminate y. The y terms are 2y and -7y. The LCM is 14. Multiply the first equation by 7 and the second equation by 2. 7(5x + 2y) = -9(7) 2(12x 7y) = 2(2) 35x + 14y = x 14y = 4 Add them together 59x = -59 x = -1 Solve for y by substituting x=2 into y = -3x + 5. y = -3(2) + 5 = = -1 Therefore solution is (2,-1) CHECK by substituting solution into the other equation and see if it is true. 3x + y = 5 3(2) + (-1) =5 5 = 5 YES! Solve for y by substituting x=-1 into either of the two original equations. 5(-1) + 2y = -9 2y = -4 y = -2 Therefore solution is (-1,-2) CHECK by substituting solution into the other equation and see if it is true. 12 x 7y = 2 12(-1) 7(-2) = = 2 2 =2 YES!

3 LINEAR INEQUALITIES To graph a LINEAR INEQUALITY, First rewrite the inequality to solve for y. If the resulting inequality is y >., Then make a dashed line and shade the area ABOVE the line. If the resulting inequality is y <.., Then make a dashed line and shade the area BELOW the line. If the resulting inequality is y.., Then make a SOLID line and shade the area ABOVE the line. If the resulting inequality is y., Then make a SOLID line and shade the area BELOW the line. If when isolating y, you must divide both sides of inequality by a negative number, Then the inequality sign must be SWITCHED. Example: -3y < 12 y > 4 Graph the solution set of 3x 2y 12 3x 12 3x 12 when dividing by a negative number, SWITCH the inequality sign 3x y x

4 WORD PROBLEMS Setting up word problems: 1) Find out what you are being asked to find. Set a variable to this unknown quantity. Make sure you know the units of this unknown (miles?, hours? ounces?) 2) If there is another unknown quantity, use the given information to put that unknown quantity in terms of the variable you have chosen. If not enough information is given, use another variable. 3) Set up a table with a row for each unknown and columns made up of the terms of one of equations above (Rate*Time=Distance, Amt*Unit Cost=Value, etc..) 4) Use the given information to combine the equations of each row of the table into equations. You need at least one equation for every variable you need to solve for. (One variable needs 1 equation, Two variables needs two equations, etc..) 5) Once one variable is solved for, you can find the other unknown. 6) Check your equation by plugging in your value for x and seeing if your equation is true. Then state your conclusion in a complete sentence.

5 Example: A seaplane flying with thewind flew from an ocean port to a lake, a istance of 240 miles, in 2 hours. Flying against the wind, it make the trip from the lake to the ocean port in 3 hours. Find the rate of the plane in calm air and the rate of the wind. Step 1) What are we asked to find? The rate of the plane in calm air and the rate of the wind. (Miles per Hour) 2) Let p = rate of the plane in calm air and w = rate of the wind. 3) The trip to the lake and back are the SAME DISTANCE. So both rows are equal to the distance given, 240 miles. Remember that when you fly with the wind, you actual rate is the rate of the plane PLUS the rate of the wind. When you fly against the wind, your actual rate is the rate of the plane MINUS the rate of the wind. Table: 4) Equations from the table. 2p + 2w = 240 Rate Time Distance = rate*time With the wind p+w 2 hrs 240 miles Against the wind p-w 3 hrs 240 miles 3p 3w = 240 Solve using addition method. Pick a variable to eliminate then multiply each equation but the number necessary to get OPPOSITE LCMs of the coefficients of that variable. If we eliminate w, the LCM of 2w and -3w is 6. So our goal is to have w s coefficients to be 6 and -6. 3(2p + 2w) = 3(240) 6p + 6w = 720 2(3p 3w) = 2(240) 6p - 6w= p = 1200 p = 100 5) Solve for the other unknown using one of the equations. 2(100) + 2w = w = 240 2w = 40 w =20 6) Check p=100 and w = 20 using the other equation to see if you are correct. 3(100) 3(20) =? = 240 yes CONCLUSION: The rate of the plane in calm air is 100mph and the rate of the wind is 20mph