1. 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
y = 1
Solution:
(1,1)
4x = 4
4 4
x = 1

2. 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

3. 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

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

5. 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

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

7. 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

8. -.05( x + y = 60 ) .05x + .02y = 1.8 -.05x - .05y = -3 - .03y = -1.2
y = 40
x + y = 60
So we would use 20 mL of 5% acid and 40 mL of 3% acid
x + 40 = 60
x = 20