Transcript Ch. 3-2
3.2 WARM - UP Solve the system graphically. 4x – 2y = -8 x+y=1 5 4x – 2y = -8 4 -4x -4x -2y = -4x – 8 -2 -2 3 y = 2x + 4 2 1 –5 –4 –3 –2 x+y=1 -x -x y = -x + 1 –1 –1 –2 –3 Solution: (-1, 2) –4 –5 1 2 3 4 5 3.2 Solving Systems Algebraically State Standard – 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, linear combination, with graphs, or with matrices. The Substitution Method 1) Solve one of the equations for a variable. 2) Substitute step 1 into the other equation. 3) Solve for the variable 4) Substitute the value in Step 2 into one of the original equations to get the other variable Extra Example 1 Solve using the Substitution method: x – 2y = 3 3(2y + 3) + 2y = 1 3x + 2y = 1 6y + 9 + 2y = 1 8y + 9 = 1 -9 -9 x –2y = 3 +2y +2y 8y = -8 x = 2y + 3 y = -1 ( 1 , -1 ) x –2(-1) = 3 x+2=3 x=1 3.2 Solving Systems Algebraically The Elimination Method 1) Multiply one or both of the equations by a constant. 2) Add the revised equations in order to eliminate one of the variables. 3) Substitute the value in Step 2 into one of the original equations to get the other variable Extra Example 3a Solve using the Elimination method: 2x – 4y = 13 4x – 5y = 8 2x – 4(-6) = 13 -2(2x – 4y = 13 ) 4x – 5y = 8 -4x + 8y = -26 4x – 5y = 8 3y = -18 y = -6 2x + 24 = 13 - 24 -24 ( -11 2 , -6 ) 2x = -11 x = -11 2 Extra Example 3b Solve using the Linear Combination method: 2x + 3y = -1 -5x + 5y = 15 2x + 3 (1) = -1 5( 2x + 3y = -1) 2(-5x + 5y = 15 ) 10x + 15y = -5 -10x + 10y = 30 25y = 25 y=1 2x + 3 = -1 - 3 -3 ( -2 , 1 ) 2x = -4 x = -2 If you get the variables to cancel and you get: 0=0 You will have: Infinitely many solutions If you get the variables to cancel and you get: 0 = (some #) You will have: No solutions Example 3 A caterer is planning a party for 64 people. The customer has $150 to spend. A $39 pan of pasta feeds 14 people and a $12 sandwich tray feeds 6 people. How many pans of pasta and how many sandwich trays should the caterer make? 14(2)+ 6s = 64 -2( 14p + 6s = 64) 39p + 12s = 150 -28p – 12s = -128 39p + 12s = 150 11p 28 + 6s = 64 - 28 -28 6s = 36 s=6 = 22 p=2 The caterer should make 2 pans of pasta and 6 trays of sandwiches. Guided Practice 6x + 6y = 3 x – 6y = 6 4x + 4y = 2 -3x + 2y = -2 -5x + 7y = 10 3x – 3y = 3 15x – 21y = 22 -4x + y = 21 -2x + y = 13 -4x + 8y = 24 x – 4y = -31 -x + 2y = 6 HOMEWORK Due Tomorrow: pg. 130 – 131 (1 – 13) eoo, (19 – 39) eoo, (54 – 59) all