SECTION 22.1: A QUICK REVIEW OF THE FORMS OF A LINE
SlopeIntercept Form Of A Line
y = mx + b,
where m and b are real numbers
m is the slope of the line and b is the yintercept 
ex. y = 2x + 4
m, the slope, is 2, and b, the yintercept, is 4.
ex. 2) y = 3x  7 = 3x + 7
m, the slope, is 3, and b, the yintercept, is 7.
Please note that in order to get the correct values for b, the y intercept,
the terms of the equation must be ADDED. 
Standard Form, also called General Form, for the Equation of a Line
ax + by = c or ax + by + c = 0 where a, b amd c are integers and a is nonnegative. 
ex. 1) 3x + 2y = 5 where a = 3, b = 2 and c = 5
ex. 2) 7x  3y = 2
Can be rewritten as: 7x + 3y = 2
where a = 7, b = 3 and c = 2
ex. 3) 9x  6y  8 = 0
Can be rewritten as: 9x + 6y + 8 = 0
where a = 9, b = 6 and c = 8
Please note that in order to get the correct values for a, b and c
the terms of the equation must be ADDED.

Point Slope Form for the Equation of a Line
y  y_{1} = m(x  x_{1} ) where m is the slope of the line
and (x_{1 }, y_{1} ) is a point on the line and m, x_{1 }and y_{1} are real numbers 
ex. 1) y  3 = 4(x  7)
The slope, m, is 4.
The point
(x_{1 }, y_{1} ) is (3, 7)
ex. 2) y + 8 = 9( x  5)
Can be rewritten as: y  8 = 9( x  5)
The slope, m, is 9.
The point (x_{1 }, y_{1} ) is (8, 5)
ex. 2) y + 2 = 13( x + 7)
Can be rewritten as: y  2 = 13( x  7)
The slope, m, is 13.
The point (x_{1 }, y_{1} ) is (2, 2)
Please note that in order to get the correct values for (x_{1 }, y_{1} )
you must write the operations as subtraction. 
SECTION 22.1: WHAT IS A SYSTEM OF EQUATIONS? AND, WHAT IS SOLVING A SYSTEM OF EQUATIONS?
A "system of equations" is a collection of two or more equations with the same "unknowns." The "unknowns" are the variables in the given equations.
2 Examples of System of Equations WithTwo Unknowns x And y
ex. 1) y = 3x + 7 and y = 2x  1
ex. 2) 2x + 3y  1 = 0 and 20x + 30y  10 = 0 
You will be asked to "solve a system of equations." When you are asked to "solve a system of equations" the goal is find numbers for each of the unknowns that will make every equation in the collection(system) TRUE.
I will be examining Systems of Equations that produce lines. It can be shown that a system of equations that produces lines can have
 ONE SOLUTION
...
This is where the lines intersect on a graph. The slopes of the lines will be different.
 NO SOLUTION
...This means the two lines are parallel and never intersect. The slopes of the lines are the same but the lines have different y intercepts.
 INFINITE SOLUTIONS
...This means that one lines goes over the other line and intersect at all points. The slopes and yintercepts of the lines are the same.
ex. 1) y = 3x + 7 and y = 2x  1
Here is an example of a "systems of equations" in slope intercept form.
In this sysem of equations
there are two unknowns, x and y.
The solution is (8, 17) where x = 8 and y = 17.
TEST y = 3x + 7 FOR TRUTH


TEST y = 2x  1 FOR TRUTH 
I'll replace y by 17 and x by 8 

I'll replace y by 17 and x by 8 
17 ?=? 3(8) + 7
17 ?=? 24 + 7
17 = 17 

17 ?=? 2(8)  1
17 ?=?16  1
17 = 17 
TRUE 

TRUE 
Since (8, 17) makes both equations true then it is said that (8, 17) is a solution for the
sysyem of equations y = 3x + 7 and y = 2x  1.
Here are the graphs of the two equations. Notice that the solution (8, 17) is also where the two lines intersect. (8, 17) IS WHERE THE TWO LINES ARE EQUAL.

ex. 2) y = 2x + 1 and y = 2x  1
Here is an example of a "systems of equations" in slope intercept form.
In this sysem of equations
there are two unknowns, x and y.
There is NO SOLUTION since the slopes are the same and the y intercepts are different. These equations will produce parallel lines. See below.
ex. 3) 2x + 3y  1 = 0 and 20x + 30y  10 = 0
Here is an example of a "systems of equations" in standard form
In this sysem of equations
there are two unknowns, x and y.
I will reconstruct each of these equations into standard form so I can know there slopes and yintercepts.
2x + 3y  1 = 0


20x + 30y  10 = 0 
3y = 2x + 1 

30y = 20x + 10 
y = (2x + 1)/3 

y = (20x + 10)/30 
y = (2/3)x + 1/3 

y = (20/30)x + 10/30 = (2/3)x + 1/3 
Since both equations are equivalent and reproduce the same equation of a line in slope intercept form then the two lines intersect everyplace and have an infinite number of solutions. You would only see one line in your graph.


SECTION 22.2: HOW TO SOLVE A SYSTEM OF TWO
LINEAR EQUATIONS EQUATIONS
I will give you two equations, in this case linear, each having two variables
x and y. I will then find an ordered pair, (x, y), that will makes both
equations true.
HOW TO SOLVE A SYSTEM OF EQUATIONS BY SUBSTITUTION 
Step 1) Solve for one of the variables in one of the equations, generally y.
Step 2) Take the result of the solution and substitute it into the same variable of the other equation
Step 3) Solve the resulting equation you get in step 2)
Step 4) Take the result from step 3) and substiture that result into the proper variable of either one of the two original equations.
Step 5) You will now have the ordered pair that will be a solution to the system of equations.

ex. 1) Given x + y = 12 and x  y = 6, solve the systems of equations.
Step 1) I solved for y in x + y = 12.
Step 2) x  y = 6 → x  (12  x) = 6 I substituted y = 12  x into y of x  y = 6
Step 3) x  (12  x) = 6 → x  12 + x = 6 → 2x = 6 + 12 → 2x = 18 → x = 9
...I solved for x.
Step 4) Using x + y = 12, one of the two original equations, I'll substitute x = 9 into the variable x in x + y = 12.
x = 9: 9 + y = 12 → y = 12  9 → y = 3
Step 5) (9, 3) is the ordered pair that will make both x + y = 12 and x  y =
6 true!
Let's test the solution in both x + y = 12 and x  y = 6.
Place (9, 3) into x + y = 12: 9 + 3 = 12 is true!
Place (9, 3) into x  y = 6: 9
 3 = 6 is true!
SECTION 22.3: SOLVING A SYSTEM OF TWO LINEAR EQUATIONS AND TWO
UNKNOWNS GRAPHICALLY
I found the solution that made the system of equations, x + y = 12 and x  y = 6, true in the previous section. I propose that if I graph the system of linear equations x + y = 12 and x  y = 6, then those two lines will intersect at (9,
3), which is the solution I got in the previous section using the substituion technique.
Let's see!
Solve Equation 1 for y to prepare for graphing: x + y = 12 → y = x + 12
Solve Equation 2 for
y: x  y = 6 → y = x + 6 → (1)(y) = (1)(x  6) → y = x + 6
Graph both equations, y = x + 12 and y = x + 6, on the same graph. I am assuming that you can graph these equations of lines. Notice that the lines intersect at (9, 3).
ex 2) Solve the system of linear equations 3x  y = 1 and x + y = 7 by substitution and graphing
Solving by Substitution
3x  y = 1 → y = 3x + 1 → (1)(y) = (1)(3x + 1) → y = 3x  1
I'll substitute 3x  1 into y of the other linear equation x + y = 7.
So: x + (3x  1) = 7 → x + 3x 1 = 7 → 4x = 7 + 1 → 4x = 8 → x = 2
Now substitute x = 2 into either one of the original equations to get the y
value:
I use 3x  y = 1: 3(2)  y =1 → 6  y = 1 → y = 5 → y = 5
Therefore, the solution is (2, 5) and will make both 3x  y = 1 and x + y = 7 TRUE!
Solving by Graphing
I need to solve for y in both 3x  y = 1 and x + y = 7.
I already solved for y using 3x  y = 1 above. I got y = 3x  1.
For x + y = 7: y = x + 7
I will now graph y = 3x  1 and y = x + 7 and see where they intersect.
The point of intersection at (2, 5) is the same ordered pair that we
got from "Solving the System of Equations by Substitution!
 QUESTION 
WHICH TECHNIQUE, BY GRAPHING OR BY SUBSTITUTION
WOULD BE ORE ACCURATE?
WHY?

SECTION 22.4: MORE TECHNIQUES FOR SOLVING SYSTEMS OF EQUATIONS
There are other techniques for solving systems of equations. One such technique is called the "Addition / Subtraction" Technique. Another technique is using matrices and "Cramer's Rule." Feel free to research those techniques and see if find these techniques easier for you to use.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXTRA CREDIT IF YOU COME IN KNOWING AND
SHOWING THESE TECHNIQUES TO THE CLASS
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!



THE SUMMER STUDY GUIDE
BY CHAPTERS

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
 CHAPTER 1: THE NUMBER SYSTEM
 CHAPTER 2: ORDER OF OPERATIONS
 CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
 CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
 CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
 CHAPTER 6: PROPERTIES OF NUMBERS
 CHAPTER 7: THE WORLD OF FRACTIONS
 CHAPTER 8: EXPONENTS
 CHAPTER 9: ROOTS
 CHAPTER 10: ALGEBRAIC EXPRESSIONS
 CHAPTER 11: CARTESIAN COORDINATE SYSTEM
 CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
 CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
 CHAPTER 14: XINTERCEPT(ZERO) AND Y INTERCEPT(B)
 CHAPTER 15: LINES
 CHAPTER 16: FUNCTIONS
 CHAPTER 17: MULTIPLYING POLYNOMIALS
 CHAPTER 18: FACTORING
 CHAPTER 19: RATIONAL EXPRESSIONS
 CHAPTER 20: SOLVING EQUATIONS
 CHAPTER 21:SOLVING INEQUALITIES
 CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
 CHAPTER 23: QUADRATICS
 CHAPTER 24: CIRCLES
 CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
 CHAPTER 26: VOLUMES
