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## Algebra

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**Algebra**7.3 Solving Linear Systems by Linear Combinations**This is the third and final way to solve linear systems.**• The other two are ____________ and ______________. graphing substitution**Steps**• Arrange the equations with like terms in columns. • Multiply one or both equations by a number to obtain coefficients that are opposites for one variable. • Add the equations. One variable will be eliminated. Solve for the other. • Substitute this number into either original equation and solve for the other variable. • Check.**-2x + 2y = -8**2x + 6y = -16 8y = -24 y = -3 2x + 6y = -16 2x + 6(-3) = -16 2x – 18 = -16 2x = 2 x = 1 Solution: (1, -3) Solve Check: -2(1) + 2(-3) = -8 2(1) + 6(-3) = -16**3x = -6y + 12**-x + 3y = 6 Rewrite the top: 3x + 6y = 12 -x + 3y = 6 -3x + 9y = 18 15y = 30 y = 2 -x + 3y = 6 -x + 3(2) = 6 -x + 6 = 6 -x = 0 x = 0 Solution: (0, 2) Solve [ ]3 Check: 3(0) = -6(2) + 12 -(0) + 3(2) = 6**3x + 5y = 6**-4x + 2y = 5 -12x + 6y = 15 26y = 39 y = 39/26 y = 3/2 -4x + 2(3/2) = 5 -4x + 3 = 5 -4x = 2 x = -½ Answer: (-½, 3/2) Solve 12x + 20y = 24 [ ]4 [ ]3 Check: 12(-½) + 20(3/2) = 24 -4(-½) + 2(3/2) = 5**2x + 8y = -2**5x + 4y = 3 -10x - 8y = -6 -8x = -8 x = 1 2(1) + 8y = -2 2 + 8y = -2 8y = -4 y = -½ Answer: (1, -½) You try! Solve. [ ]-2 Check: 2(1) + 8(-½)= -2 5(1) + 4(-½) = 3**A boat traveled from 24 miles downstream in 4 hours. It**took the boat 12 hours to return upstream. Find the speed of the boat in still water(B) and the speed of the current(C). Speed in still water + current speed = speed downstream Speed in still water – current speed = speed upstream B + C = 6 mph B – C = 2 mph 2B = 8 mph B = 4 mph 4 mph + C = 6 mph C = 2 mph The boat goes 4 mph. The current goes 2 mph. Speed downstream is 24 miles/4 hours = 6 mph Speed upstream is 24 miles/12 hours = 2 mph**HW**P. 414-415 (#9-41 4X) (#45-48)