Solving Systems of Linear Equations by Graphing

Systems of Linear Equations Lesson #3: Solving Systems of Linear Equations by Inspection and by Substitution Method of Inspection In some simple cases, a system of linear equations can be solved by mentally trying different values for the variables until a correct solution is reached. This is called the method of inspection and is really only practical if the equations are very simple. Claus Ex. # The sum of two numbers is 14 and the difference between the numbers is 2. Form two equations in two variables and determine the numbers by inspection. Let the larger nuriber, Let y the smaller number x+y=147 while adding towo numbers which equal to 14, inspect the same x-y2 two numbers and see it they subtract to 2. (3)円 13-112 от 2 10-4-19 10-4-6 VTNET 2 86-14-6-2 by inspection, x=8 and y=6 Class Ex. 02 Solve the systemx+2y=12 andx+3y=17 by inspection. Interpret the two equations in words 1) x+2y=12 11) x+3y=17 "a number plus trice another number equals 12" 4"a number plus three times another number equals 19" i) avess and check numbers in x+2y=12, then try thon in 2-3y 17 x+2y=12 sumbers should be whol becuese inspection is only pratical when equations are simple. Try x=2 (2)+2y-12 2y10 x+3y=17 (2)+3(5) 17 17=17 Works! x=2 and y=5 Try x=2 and 4-5 in other equation Complete Assignment Questions #1-#3 466 Systems of Linear Equations Lesson #3: Solving Linear Systems by Inspection and by Substitution Method of Substitution If the equations are too complex to be solved by inspection, then algebraic procedures such as the method of substitution and the method of elimination (next lesson) may be used. When using the method of substitution, there are four steps which are shown in the flowchart below. Step 1 Choose the simpler equation and express one variable in terms of the other. Step 2 Substitute the expression from step 1 into the other equation. Step 3 Solve the single variable equation. Step 4 Substitute the solution from step 3 into the equation in step 1 to find the value of the other variable. Claas Ex. 43 Consider the following system of equations: x+4y=17 2x-y-7 a) Solve the system using the method of substitution by rewriting the first equation in the form x= x+4y=17 2x-y=7 x=-44+17 x+4y=17 x+4(8)=17 2+12=17 x=5 72(-4y+19)-y-7 colve Sy 34-4-7 -=-27 y=3 for Piva into x=5 y=3 Plug y valve found into the other equation solve tur x. b) Solve the system using the method of substitution by rewriting the first equation in the form y= 2x-y=7 x+4y=17 49=-x+17 2x-(x)=7 y=-x+(2x+x)=(7) multiplying 8x+X-14-28 by 4 mes the numbres easier to work with! No fractions! 9x = 45 y=-x+ y=(3)+ y=3 e) Which method, a) or b), was simpler? x=5 d) Verify that the solution satisfies both equations. x+4y = 17 LS: (S)+4(8):17 RS: 17 2x-y-7 LS: 2(5)-(3)=7 PS: 7 LS-PS LS-RS verified e) Check the solution using a graphing calculator. x=5 y=3 Systems of Linear Equations Lesson #3. Solving Linear Systems by Inspection and by Substitution 467 CM Consider the following system of equations: 4x+3y=0, 8x-9y5. a) Solve and verify the system using the method of substitution. 4x+3y=0 8x-4y=5 y=x 3y=-x =(4) y=-x 8x-9(-x)=5 8x+12x5 20x=5 x= y= 3 x= y= -1 verity 4x+3y0 LS: 46)-3(1)=0 RSO 8x-94-5 25:8(+)-9(+)-5 LS-PS LS = RS verified✓ CsEx. 45 b) Check the solution using a graphing calculator. Consider the following system of equations: 5(2a-3)+b=5, 6а-2(b-4)=20. a) Solve the system using the method of substitution. 5(2a-3)+b-5 69-2(0-4)-20 b=-100-20 104-15-6-5 b=-10(2)+20 104+b=20 69-21-109-20-4)=20 69+209-40+8=20 269=52 a=2 b=0 b=-109-20 b-0 b) Verify algebraically that the solution satisfies both equations. 5(29-3)+b=5 64-2(6-4)=20 25:5(2(2)-3)+(0) = 5 28:6(2)-2(10)-4) = 12+8=20 RS: 5 PS: 20 LS=RS LS = RS remfied Complete Assignment Questions #4 - #10