Chapter 2
Solving Linear Equations
Mathematically Speaking
Can you identify what happens in each step?
15x + 13y – 4(3x+2y)
15x + 13y – 12x - 8y
15x – 12x + 13y - 8y
(15 – 12)x + (13 – 8)y
3x + 5y
Can you identify what has happened
in each step?
- Given
15x + 13y – 4(3x+2y)
15x + 13y – 12x - 8y -Distributive
15x – 12x + 13y - 8y -Commutative
-Factor
(15 – 12)x + (13 – 8)y
-Addition
3x + 5y
Identify the steps used to solve the
equation, m + 4 = 29.
m+4=29
- 4=-4
m =25
Given
Inverse +  Evaluate
Identify the steps used to solve the
equation.
3x + 4 = 19
-4=-4
3x = 15
3= 3
x= 5
Given
Inverse +  Evaluate
Inverse *  
Evaluate
Identify the steps used to solve the
equation.
5x – 4 = 2(x – 4) + 18
5x – 4 = 2x – 8 + 18
5x – 4 = 2x + 10
3x = 14
14
x
3
Identify the steps used to solve the
equation.
5x – 4 = 2(x – 4) + 18
5x – 4 = 2x – 8 + 18
5x – 4 = 2x + 10
3x = 14
14
x
3
•
Given
Distributive
Addition
Inverse Ops
Like terms
Inverse Ops
Identify the steps used to solve the
equation.
-5x + 3 + 2x = 7x – 8 + 9x
-3x +3 = 16x -8
11 = 19x
11
x
19
11
x
19
Identify the steps used to solve
the equation.
-5x + 3 + 2x = 7x – 8 + 9x Given
-3x +3 = 16x -8 Like Terms
11 = 19x
Inverse Ops
11
Inverse Ops
x
19
Symmetric
11
x
19
property
So what is the definition? Which of
these equations are linear?
Not Linear
Linear
x+y = 5
2x+ 3y = 4
7x-3y = 14
y = 2x-2
3
y=4
x2 + y = 5
x=5
y
xy = 5
x2 +y2 = 9
y = x2
The degree must be one.
2.1 What is a solution?
What happens when one solves an
equation?
You might say “One gets an answer.”
What is the format of that answer?
What happens when one solves an
equation?
1. The solutions is a Unique solution.
2. The solution is Infinite solutions.
3. The is no possible solution.
What happens when one solves an
equation?
1. The solution is a Unique solution.
•
There is only ONE numerical answer to
solve the equation.
2. The solution is Infinite solutions.
•
IDENTITY. The equations are
mathematically equivalent.
3. There is no possible solution.
•
INCONSISTENT. With linear equations
this means there is no point of
intersection.
2.2 One linear equation in one
variable
One Solution.
3x + 4 = 19
-4=-4
3x = 15
3= 3
x= 5
Infinite Solutions. IDENTITY
14 + 5x – 4 = (x + 4x)-8 +
18
14 + 5x – 4 = 5x – 8 + 18
5x + 10 = 5x + 10
10 = 10
No Solution. INCONSISTENT
-7x + 3 + 1x = 2x – 8 - 8x
-6x +3 = -6x -8
3 = -8
3  -8
2.3 Several linear equations in
one variable
Systems of Equations
Solving systems of
equations with two or more
linear equations
Substitution
Elimination
Cramer’s Rule
Graphical Representation
The 3 possible solutions still occur.
1. The solution is a Unique solution.
•
This one solution is in the form of a
point. (e.g. (x,y), (x,y,z) )
2. The solution is Infinite solutions.
•
IDENTITY. The lines are the same line.
3. There is no possible solution.
•
INCONSISTENT. The lines are
parallel (2-D) or skew (3-D).
Substitution – use substitution
when…
One of the equations is already solved for a
variable.
y = 2x – 5
3x + 4y = 13
Substitute the first equation into the second
3x + 4(2x – 5) = 13
Solve for the variable
3x + 8x – 20 = 13
11x = 33
x=3
Substitute back into one of the original equations
y = 2(3) – 5 = 1
Final Answer (3,1)
Elimination – use elimination when
substitution is not set up.
Elimination ELIMINATES a variable through
manipulating the equations.
Some equations are setup to eliminate.
Some systems only one equation must be
manipulated
Some systems both equations must be
manipulated
Setup to Eliminate
Given
2x – 4y = 8
3x + 4y = 2
The y terms are opposites, they will eliminate
Add the two equations
5x = 10  x = 2
Substitute into an original equation
3(2) + 4y = 2  6 + 4y = 2 
4y = -4  y = -1
Final Answer (2,-1)
Manipulate ONE eqn. to
Eliminate
Given
2x + 2y = 8
3x + 4y = 2
Multiply the first equation by – 2 to elim. y terms
-4x – 4y = -16
3x + 4y = 2
Add the two equations
-1x = -14  x = 14
Substitute into an original equation
3(14) + 4y = 2  42 + 4y = 2 
4y = -40  y = -10 Final Answer (14,-10)
Manipulate BOTH eqns. to Eliminate
Given
2x + 3y = 4
3x + 4y = 2
Multiply the first equation by 3 & the second
equation by -2 to elim. x terms
6x + 9y = 12
-6x - 8y = -4
Add the two equations
y=8
Substitute into an original equation
2x + 3(8) = 4  2x + 24 = 4 
2x = -20  x = -10
Final Answer (-10,8)
Identity Example
2x + 3y = 12
y = -2/3 x + 4
Using substitution
2x + 3(-2/3 x + 4) = 12
2x – 2x + 12 = 12
12 = 12
Identity
Inconsistent Example
3x – 4y = 18
3x – 4y = 9
Use Elimination by multiplying Eqn 2 by
-1.
3x – 4y = 18
-3x + 4y = -9
0 = 9 False
Inconsistent
3 Equations: 3 Variables required
Eqn1: 3y – 2z = 6
Eqn2: 2x + z = 5
Eqn3: x + 2y = 8
Solve Eqn2 for z
z = -2x + 5
Now substitute into Eqn1
3y – 2(-2x+5) = 6
3y + 4x – 10 = 6
3 Equation continued…
NEW: 4x + 3y = 16
Eqn3: x + 2y = 8
Now one can either substitute or eliminate
NEW: 4x + 3y = 16
Eqn3(*-4): -4x - 8y = -32
-5y = -16
y = 16/5
And still continued…
Now having a value for y, one can
substitute into
x + 2(-16/5) = 8
x = 8 + 32/5 = 40/5 + 32/5
x = 72/5
This can now be substituted into our
Eqn2 solved for z
z = - 2(72/5) + 5
z = -144/5 + 5 = -144/5 + 25/5
z = -119/5
Final Answer
(72/5, -16/5, -119/5)
Matrices: Cramer’s Rule
Dimensions: row x columns
a
c
b
d
Determinant
ad - bc
e
f
Cramer’s Rule set up
e
x= f
b
d
determinant
y=
a
b
e
f
determinant
Example
2x + 3y = 5
4x + 5y = 7
2
4
3
5
5
7
The determinant is 10-12 = -2
x=
5
7
3
5
-2
y=
2
4
5
7
-2
Solve for x and y…
5 3
2
x setup
y setup
7 5
4
-2
25 - 21
-2
4/-2
x = -2
5
7
-2
14 - 20
-2
-6/-2
y=3
Final answer (-2,3)
You cannot use Cramer’s Rule if
the difference of the products is 0.
Verbal Models
Verbal Models are math problems
written in word form
General Rule: Like reading English Left to Right
Special Cases: Change in order terms
some time called “turnaround” words
(Cliff Notes: Math Word Problems,
2004)
Convert into Math…
Two plus some
number
A number decreased
by three
Nine into
into thirty-six
Seven cubed
Eight times a number
Ten more
more than
than five is
what number
2+x
x-3
36 / 9
7^3 73
8x
5 + 10 = x
MORE Convert into Math…
Twenty-five percent of
what number is twentytwo?
The quantity of three
times a number
divided by seven
equals nine.
The sum of two
consecutive integer is
23.
.25 * x = 22
(3x)/7 = 9
x + (x+1) = 23
Work Problem.
I can mow the yard in 5 hours. My
husband can mow the yard in 2
hours. If we mowed together how long
would it take for us to mow the yard.
Solution
My rate is 1 yard per 5 hours: 1/5 t
Doug’s rate is 1 yard per 2 hours; ½ t
together = addition
The whole job = 1
1 1
t  t 1
5
2
the common denominator is 10
2t  5t  10
Solve for t
7t = 10; t = 10/7 or 1.42857 ish
Formulas you should know…
Area of
Rectangle
Perimeter of
Rectangle
Area of Triangle
Area of Circle
A = hb
P = 2 (h + b)
A = ½ hb
A = pr2
Candy
I bought 3 bags of candy and 5
chocolate bars. I spent $13. My friend
spent $17 and she bought 4 bags of
candy and 6 chocolate bars. What is
the cost of the candy bags and
chocolate bars?
Solution
3b + 5c = 13
4b + 6c = 17
det = 18-20 = -2
x = 13 5
17 6
-2
x = (78-85)/-2
x = -7/2 = $3.50
3
4
5
6
y=
3
4
13
17
13
17
-2
y = (51-52)/-2
y = -1/-2 = 0.50