Chapter+7

= **7.1** =

** Graphing systems of equations **
__ Example: __ Equation 1: y = 3x Equation 2: 4x + 2y = 30

***The point that the equations intersect is the answer __ Steps __ ............. Yes - this is the answer ............. No - if they are parallel they have no solution (null) .................. - if its the same line (infinite solution)
 * 1) Graph the first equation
 * 2) Graph the second equation
 * 3) Do they intersect?

__**Systems of Equations**__
 * **Graph of a System** || Intersecting Line || Same Line || Parallel Line ||
 * **Number of Solutions ... ** || Exactly one solution || Infinitely many || No solutions ||
 * **Terminology** || Consistent/independent** ... ** || Consistent/independent** ... ** || Inconsistent ||

= **7.2** =

**Substitution**
__Steps__ 1. (Second equation) 2. Substitute 3. Simplify 4. Combine like terms 5. Get //x// by itself 6. Simplify 7. (First equation) 8. (Repeat steps 2-6 to get the value of //y//)

Two numbers not equal = no solution Two numbers are equal = infinite solutions

**Ex:** 2x + 4 = y ...... x = 3

2(3) + 4 = y 6 + 4 = y  y = 10 (3, 10)

Works best when you have one of your variables alone!** = = = **7.3** =

If the signs are the same, you need to subtract If the signs are different you need to add

***Addition**
-3x + 4y = 12 ........ -6y + 3x = 18

**1.)** Rearrange the terms. -3x + 4y = 12  __ 3x + -6y = 18 __  ...  **2.)** Add the terms together. -3x + 4y = 12 __3x + -6y = 18__ 0x + -2y = 30 ... **3.)** Solve for the variable.  y = -15  ...  **4.)**** Substitute the variable back. -3x + 4(-15) = 12 -3x + -60 = 12 -3x + -60 + 60 = 12 + 60 -3x = 72 x = -24 so ( -24, -15) is the answer

***Subtraction**
3z - 8w = -183 ........ z + 9 = -4w

** 1.) ** Rearrange the terms.. 3z – 8w = -18  __ 3z + 4w = -9 __

** 2.) ** Subtract the terms. 3z – 8w = -18  __(-) 3z + - 4w = --9__  0z + -12w = -9

** 3.) ** Solve for the variable. __-12w__ = __-9__  -12 .... -12  w = 3/4

** 4.) **** Substitute the variable back. 3z - 8(3/4) = -18  3z – 6 = -18  3z – 6 + 6 = -18 + 6  3z = -12 z = -4 so (3/4, -4 )

= **7.4** =

** Elimination using multiplication **
-Multiply one equation by whatever you need to in order to eliminate one variable -Solve to find the value of the other -Insert your value into one of the original equations -Solve to find the value of the remaining variable.

= **7.5** =

** Graphing inequalities **
__Steps:__ 1. Graph first inequality 2. Graph second inequality 3. Find the part where they over lap (your answer)