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SOML MEET 4 NAME: ___________________ EVENT 1 TEAM: ___________________ SYSTEMS OF LINEAR EQUATIONS SCHOOL: ___________________ 1. [2 Points] Eight times a certain number added to five times a second number is 184. The first number minus the second number is -3. Find the numbers. ANS: ____________________ 2. [3 Points] A landscaper has two kinds of solutions containing weed killer and water. One is 5% weed killer and the other is 15% weed killer. The landscaper needs 100 liters of a 12% solution and wants to make it by mixing the two solutions. How much of each solution should be used? ANS: ________ L of 5% ________ L of 15% 3. [5 Points] Solve the following system of equations: 3x + 2y - z = 0 5x - y - 8z = 9 x + 4y - 3z = -22 ANS: x = ______ y = ______ z = ______ SOML MEET 4 EVENT 1 SYSTEMS OF LINEAR EQUATIONS 1. [2 Points] NAME: _Key_________ Eight times a certain number added to five times a second number is 184. The first number minus the second number is -3. Find the numbers. Solution: Let x = the first number and y = the second number 8x + 5y = 184 x - y = -3 ← multiply through by 5 8x + 5y = 184 5x - 5y = -15 Now add these two equations together: 13x = 169 x = 169/13 = 13 Now substitute x = 13 into: x - y = -3 13 - y = -3 13 + 3 = y 16 = y The first number is 13. The second number is 16. ANS: ___13, 16_______ 2. [3 Points] A landscaper has two kinds of solutions containing weed killer and water. One is 5% weed killer and the other is 15% weed killer. The landscaper needs 100 liters of a 12% solution and wants to make it by mixing the two solutions. How much of each solution should be used? Solution: Let: x = liters of the 5% weed killer solution y = liters of the 15% weed killer solution x + y = 100 The two solutions combined will total to 100 liters 0.05x + 0.15y = 0.12(100) = 12 The two solutions combined will have a total of 12 liters of weed killer. Solve the first equation for y: y = 100 - x Substitute y with (100-x) into the second equation: 0.05x + 0.15y = 12 0.05x + 0.15 (100 - x) = 12 0.05x + 15 - 0.15x = 12 - 0.10x = -3 x=3 Plug 30 in for x: y = 100 - x = 100 - 30 = 70 We will need 30 liters of the 5% weed killer solutions and 70 liters of the 15% weed killer solution. ANS: __30____ L of 5% __70____ L of 15% 3. [5 Points] Solve the following system of equations: 3x + 2y - z = 0 5x - y - 8z = 9 x + 4y - 3z = -22 Solution: Solve the first equation for z: z = 3x + 2y Use this expression to replace z in the other two equations of the system: 5x - y - 8z = 9 5x - y - 8(3x + 2y) = 9 5x - y - 24x - 16y = 9 x + 4y - 3z = -22 x + 4y - 3(3x + 2y) = -22 x + 4y - 9x - 6y = -22 -8x - 2y = -22 ← Divide by 2 to simplify -19x - 17y = 9 -4x - y = -11 Now we have a system of two equations: -19x - 17y = 9 -4x - y = -11 Solve the second equation for y: y = -4x + 11 Substitute this into the other equation: -19x - 17(-4x + 11) = 9 -19x + 68x - 187 = 9 49x = 196 x=4 Plug x = 4 into the equation previously solved for y: y = -4x + 11 = -4(4) + 11 , so y = -5 Plug x = 4 and y = -5 into the equation previously solved for z: z = 3x + 2y z = 3(4) + 2(-5) z=2 The solutions to the system of equations are: ANS: x = ___4___ y = __-5___ z = ___2___