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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 22: SOLVING TWO LINEAR EQUATIONS AND TWO UNKNOWNS.
ALSO CALLED "SOLVING A SYSTEM OF EQUATIONS"

 

SECTION 22.1: A QUICK REVIEW OF THE FORMS OF A LINE

Slope-Intercept 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 y-intercept


ex. y = 2x + 4  
m, the slope, is 2, and b, the y-intercept, is 4.

ex. 2) y = -3x - 7 = -3x + -7
m, the slope, is -3, and b, the y-intercept, 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 non-negative.


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 -
y1 = m(x - x1 ) where m is the slope of the line
and (x1 , y1 ) is a point on the line and m, x1 and y1 are real numbers


ex. 1) y - 3 = 4(x - 7)
The slope, m, is 4.
The point (x1 , y1 ) 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 (x1 , y1 ) 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 (x1 , y1 ) is (-2, -2)

Please note that in order to get the correct values for (x1 , y1 )
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

  1. ONE SOLUTION
    ... This is where the lines intersect on a graph. The slopes of the lines will be different.


  2. 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.

  3. INFINITE SOLUTIONS
    ...This means that one lines goes over the other line and intersect at all points.
    The slopes and y-intercepts 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.

2

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.

3

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 y-intercepts.

 
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.

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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) MATH       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).


5

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.

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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?
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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: X-INTERCEPT(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