Quiz -1( 4x + 2y = 6 4x + y = 5 ) -4x - 2y = -6 4x + y = 5 -y = -1 Solve the following system using the substitution or elimination method -1( 4x + 2y = 6 4x + y = 5 ) -4x - 2y = -6 4x + y = 5 -y = -1 4x + 1 = 5 -1 -1 - 1 - 1 y = 1 Solution: (1,1) 4x = 4 4 4 x = 1

Special Types of Linear Systems 2x + y = 5 2x + y = 1 What happens if we solve this system? ) -1( 2x + y = 5 -2x - y = -1 0 = 4 The statement 0 = 4 is a false statement – that means that there is no solution. If we graphed these lines they would be parallel – they don’t cross

Special Types of Linear Systems ) -2( 2x + y = 5 4x + 2y = 10 What happens if we solve this system? -4x - 2y = -10 4x + 2y = 10 0 = 0 The statement 0 = 0 is a true statement – that means that every point is a solution. If we graphed these lines they would be the same line – they cross at every point

Practice: Solution (-3,2) x + 2y = 1 5x – 4y = -23 Solve using Substitution 5x – 4y = -23 x + 2y = 1 -2y -2y 5(1 – 2y) – 4y = -23 x = 1 – 2y 5 – 10y – 4y = -23 5 – 14y = -23 x + 2(2) = 1 -5 -5 x + 4 = 1 – 14y = -28 -4 -4 -14 -14 x = -3 y = 2

Word Problems You plant a 14-inch tree in your backyard that grows at a rate of 4 inches per year and an 8-inch tree that grows at a rate of 6 inches per year. How many years until they are the same height? 14 inches + 4(# of years) = height 8 inches + 6(# of years) = height x# of years and yheight 14 + 4x = y 8 + 6x = y We solve this system of equations

-3( 14 + 4x = y ) -42 – 12x = -3y 16 + 12x = 2y 2( ) 8 + 6x = y -1 -1 14 + 4x = y 26 = y 14 + 4x = 26 -14 -14 4x = 12 So in 3 years the plants will be the same height  26 inches tall 4 4 x = 3

Word Problems In your chemistry class you have a bottle of 5% boric acid and a bottle of 2% boric acid. You need 60 milliliters of 3% boric acid. How much of each solution should you use? mL of 5% acid + mL of 2% acid = 60mL (mL of 5%)(.05) + (mL of 2%)(.02) = 60(.03) xmL of 5% acid and ymL of 2% acid x + y = 60 .05x + .02y = (60)(.03) .05x + .02y = 1.8 We solve this system of equations

-.05( x + y = 60 ) .05x + .02y = 1.8 -.05x - .05y = -3 - .03y = -1.2 - .03 - .03 y = 40 x + y = 60 So we would use 20 mL of 5% acid and 40 mL of 3% acid x + 40 = 60 - 40 - 40 x = 20