QuizSolve the following system using the substitution or elimination method

4x + 2y = 6

4x + y = 5

-4x - 2y = -6)-1(

4x + y = 5

-y = -1 -1 -1 y = 1

4x + 1 = 5 - 1 - 1 4x = 4 4 4

x = 1

Solution:

(1,1)

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

2x + y = 5

4x + 2y = 10

What happens if we solve this system?

)-2(

-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:x + 2y = 1

5x – 4y = -23

Solve using Substitution

x + 2y = 1 -2y -2y x = 1 – 2y

5x – 4y = -23

5(1 – 2y) – 4y = -235 – 10y – 4y = -235 – 14y = -23-5 -5 – 14y = -28 -14 -14

y = 2

x + 2(2) = 1x + 4 = 1 -4 -4 x = -3

Solution

(-3,2)

Word ProblemsYou 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 yheight 14 + 4x = y 8 + 6x = y

We solve this system of equations

)-3( -42 – 12x = -3y

-26 = -y -1 -1

26 = y 14 + 4x = y 14 + 4x = 26-14 -14 4x = 12

So in 3 years the plants will be the same height 26 inches tall

14 + 4x = y 8 + 6x = y )2( 16 + 12x = 2y

4 4

x = 3

Word ProblemsIn 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)

xmL of 5% acid and ymL of 2% acid x + y = 60 .05x + .02y = (60)(.03) .05x + .02y = 1.8

We solve this system of equations

x + y = 60 .05x + .02y = 1.8

)-.05(

-.05x - .05y = -3 - .03y = -1.2 - .03 - .03

y = 40

x + y = 60 x + 40 = 60 - 40 - 40 x = 20

So we would use 20 mL of 5% acid and 40 mL of 3% acid