* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Signal-flow graph wikipedia, lookup

Quadratic equation wikipedia, lookup

Cubic function wikipedia, lookup

Quartic function wikipedia, lookup

Elementary algebra wikipedia, lookup

System of linear equations wikipedia, lookup

History of algebra wikipedia, lookup

Transcript

3.2 Solving Systems of Equations Algebraically Substitution Method Elimination Method Substitution Method Here you replace one variable with an expression. x + 4y = 26 x – 5y = - 10 Solve for a variable, x = 26 – 4y Replace “x” in the other equation (26 – 4y) – 5y = -10 Solve for y Solve for y (26 – 4y) – 5y = -10 26 – 4y – 5y = - 10 Remove parentheses by multiplying by 1 26 – 9y = - 10 Add like terms -9y = - 36 y=4 Subtract 26 from both sides Divide by - 9 Solve for x x = 26 – 4y x = 26 – 4(4) Substitute for y x = 26 – 16 x = 10 The order pair is (10, 4). This is where the lines cross. The Elimination Method Here we add the equations together when the coefficients are different signs. x + 2y = 10 x+y=6 Here both lead coefficients are 1. We can change the coefficient to – 1, by multiplying by – 1. x + 2y = 10 x+y=6 Multiply the bottom equation by – 1. x + 2y = 10 -x-y =-6 y=4 When adding the equations together, x go to zero. Find x by replace it back in either equation. x + 2(4) = 10; x + 8 = 10; x=2 So the order pair (2, 4) works in both equations. 2 + 2(4) = 10 2+4=6 We have two way to solve the systems, Substitution and Elimination; which way is better depends on the problem. What about this problem 2x + 3y = 12 5x – 2y = 11 Here we have to multiply both equations If we wanted to remove the “x”, then we have to find the Least common multiple (L.C.M.) of 2 and 5. If we wanted to remove the “y”, then we have to find the least common multiple of 3 and -2. Lets get rid of the “y” The L.C.M of 2 and 3 is 6. Since we want the coefficients to be opposite, - 2 will help in the equation. we multiply the top equation by 2. 2x + 3y = 12 4x + 6y = 24 The bottom equation by 3 5x – 2y = 11 15x – 6y = 33 Add the new equations together 4x + 6y = 24 15x – 6y = 33 19x = 57 Divide by 19 x=3 Replace in original equation and solve for y 2(3) + 3y = 12 6 + 3y = 12 3y = 6 y= 2 What about inconsistent systems? y – x = 5 Multiply the top equation by – 2, 2y – 2x = 8 2y – 2x = -10 then add the bottom. 2y – 2x = 8 0=-2 This shows no solutions. What if it is dependent (Many solutions) 1.6y = 0.4x + 1 0.4y = 0.1x + 0.25 Multiply the top and bottom equation by 100 to remove decimals. 160y = 40x + 100 40y = 10x + 25 Then multiply the bottom equation by -4 -160y = -40x – 100 Add the new equations together 160y = 40x + 100 -160y = -40x – 100 0=0 This is a system with many solutions. Solve this system a–b=2 -2a + 3b = 3 How about this system y = 3x – 4 y=4+x Homework Page 120 # 13 – 35 odd Homework Page 120 # 14 – 34 even, 37