Solving Linear Systems by Linear Combinations Objective: Students will solve a linear system by combinations ( Elimination ).

Algebra Standards: 8.EE.8 Students solve a system of two linear equations in two variables algebraically, and are able to interpret the answer graphically. Students are able to use this to solve a system of two linear inequalities in two variables, and to sketch the solution sets.

{ 3 3 -4 -4 Solve the linear system 2x + 4y = 10 Check answer! #1 Add the Equations Solve the linear system { 2x + 4y = 10 Check answer! x – 4y = 20 Solution: 2x + 4y = 10 x – 4y = 20 x – 4y = 20 10 – 4y = 20 1 3x 1 = 30 -10 -10 3 3 1 -4y = 10 x = 10 -4 -4 5 y = – 5 Solution is (10, ) – 2 2

{ 3 3 -4 -4 Solve the linear system 4x + y = -4 Check answer! #2 Add the Equations Solve the linear system { 4x + y = -4 Check answer! -4x + 2y = 16 Solution: 4x + y = -4 -4x + 2y = 16 -4x + 2y = 16 -4x +2 (4) =16 1 3y = 12 -4x + 8 =16 3 3 -8 -8 y = 4 -4x 1 = 8 -4 -4 Solution is (-2, 4) x = -2

{ -1[ ] Solve the linear system x + 5y = 2 Check answer! x – 4y = 20 #3 Multiply then add Solve the linear system { x + 5y = 2 Check answer! x – 4y = 20 Solution: x + 5y = 2 x + 5y = 2 -1[ ] x – 4y = 20 -x + 4y = -20 1 9y = -18 x + 5y = 2 9 9 y = -2 x + 5 (-2) = 2 x – 10 = 2 +10 +10 Solution is (12, -2) x = 12

{ 5[ ] 4[ ] 23 23 Solve the linear system 3x + 4y = 6 Check answer! #4 Multiply then add Solve the linear system { 3x + 4y = 6 Check answer! 2x – 5y = -19 Solution: 5[ ] 3x + 4y = 6 15x + 20y = 30 4[ ] 2x – 5y = -19 8x – 20y = -76 1 23x = -46 3x + 4y = 6 23 23 3 (-2) + 4y = 6 x = -2 -6 + 4y = 6 +6 +6 Solution is (-2, 3) 1 4y = 12 y = 3 4 4

{ -1[ ] Solve the linear system 3x + 2y = 8 Check answer! 2y = 12 – 5x #5 Move the variable first then multiply and then add Solve the linear system { 3x + 2y = 8 Check answer! 2y = 12 – 5x Solution: 3x + 2y = 8 3x + 2y = 8 2y = 12 – 5x +5x +5x -1[ ] 5x + 2y = 12 -5x – 2y = -12 1 -2x = -4 2y + 5x = 12 3x + 2y = 8 -2 -2 3 (2) + 2y = 8 x = 2 6 + 2y = 8 Solution is (2, 1) -6 -6 2y = 2 1 y = 1 2 2

-6[ ] -2 -2 In one day the National Civil Rights Museum in Memphis, #6 Word Problem In one day the National Civil Rights Museum in Memphis, Tennessee, admitted 321 adults and children and collected $1590. The price of admission is $6 for an adult and $4 for a child. How many adults and how many children were admitted to the museum that day? Solution: How many x = adults Money y = children 6x + 4y = 1590 x + y = 321 -6[ ] x + y = 321 -6x – 6y = -1926 6x + 4y = 1590 6x + 4y = 1590 – 2y 1 = -336 x + 168 = 321 -2 -2 -168 -168 153 adults y = 168 x = 153 168 children

x + y = 58 x + y = 58 x – y = 16 37 + y = 58 2x = 74 -37 -37 2 2 y = #7 Word Problem Find two numbers whose sum is 58 and whose difference is 16. Solution: x = 1st number y = 2nd number x + y = 58 x + y = 58 x – y = 16 37 + y = 58 1 2x = 74 -37 -37 2 2 y = 21 x = 37

#8 Word Problem The sum of two numbers is 56. The sum of one third of the first number and one fourth of the second number is 16. Find the numbers. x = 1st number Solution: y = 2nd number x + y = 56 -4[ ] x + y = 56 -4x – 4y = -224 1 1 12[ ] x + y = 16 4x + 3y = 192 4x + 3y = 192 3 4 – y 1 = -32 -1 -1 x +32 = 56 -32 -32 y = 32 x = 24

-3[ ] How many gallons of 30% alcohol solution and how many #9 Word Problem (Mixture) How many gallons of 30% alcohol solution  and how many of 60% alcohol solution must be mixed  to produce 18 gallons of 50% solution? Solution: Gallons of Solution x = 30% Alchol Sol. Solution(%) y = 60% Alcohol Sol. x + y = 18 10[ ] 0.3 x + 0.6 y = (0.5) 18 18 = 50% Alcohol Sol. 3x + 6y = (5) 18 -3[ ] x + y = 18 -3x – 3y = -54 3x + 6y = 90 3x + 6y = 90 1 3y = 36 x + 12 = 18 3 3 -12 -12 y = 12 x = 6

Assignment Book Pg. 290 # 1, 2, 4, 7, 19