Download Section 1.4A Notes

1. 
2. 
3. 
Describe each transformation of f to g: f(x) = |x| and g(x) = -3|x + 2| - 1. Write a function g if f(x) = x2 has a vertical shrink of 3 followed
by a translation up 2. The data shows the humerus lengths ( in centimeters) and
heights (in centimeters( of several females. Use the graphing calculator to find a line of best fit for the data. Estimate the height of a female whose humerus is 40 centimeters
long. Estimate the humerus length of a female with a height of 40
cm. Algebra II
1
Systems of Equations
with Two Variables
Algebra II
¡  two
or more linear equations.
" 1st equation
¡  Looks like #
$2nd equation
¡  A
solution is an ordered pair that makes
all equations true. €
Algebra II
3
3x – 2y = 2
x + 2y = 6
no
b) (2,2)
yes
a) (0, -1)
Algebra II
4
¡  Graphing
¡  Substitution
¡  Elimination
Algebra II
5
To find the solution of a system of two
linear equations: (steps)
1.  Graph each equation
2.  Identify the intersection
3.  This is the solution to the system
because it is the point that satisfies
both equations. **Remember that a graph is just a
picture of the solutions. Algebra II
6
Graph
Number of Solutions
Type of System
intersecting lines
one solution
Consistent
The equations are
independent.
parallel lines
no solutions
Inconsistent
The equations are
independent.
coincident lines
(same line)
infinitely many
solutions
Consistent
The equations are
dependent.
Two lines intersect at one point.
Parallel lines
Lines coincide
Algebra II
7
Solve the system of equations
by graphing.
"$ 2x − 2y = −8
#
$% 2x + 2y = 4
First, graph 2x – 2y = -8.
Second, graph 2x + 2y = 4.
The lines intersect at (-1, 3)
The solution is (-1, 3)
Algebra II
8
Solve the system of equations
by graphing.
# −x + 3y = 6
$
% 3x − 9y = 9
(3, 3)
(0, 2)
(-3, 1)
First, graph -x + 3y = 6.
€Second, graph 3x – 9y = 9.
(-3, -2)
(0, -1)
(3, 0)
The lines are parallel.
No solution
Algebra II
9
Solve the system of equations
by graphing.
# 2x − y = 6
$
% x + 3y = 10
First, graph 2x – y = 6.
€Second, graph x + 3y = 10.
The lines intersect at (4, 2)
The solution is (4, 2)
Algebra II
10
Solve the system of equations
by graphing.
# x = 3y −1
$
%2x − 6y = −2
First, graph x = 3y – 1.
€ Second, graph 2x – 6y = -2.
(-1, 0)
(2, 1)
(-4, -1)
The lines are identical.
Infinitely many
solutions
Algebra II
11
Steps for Substitution:
1.  Solve one of the equations for one variable (try to
solve for the variable with a coefficient of one)
2.  Substitute the expression into the other equation
and solve the new equation. 3.  Substitute the value from step 2 into one of your
original equations to complete the ordered pair
Algebra II
12
1. 3x – y = 6
-4x + 2y = –8
Step 2: -4x + 2y = -8
-4x + 2(3x – 6) = -8
-4x + 6x – 12 = -8
2x = 4
x = 2
Step 1: 3x – y = 6
-y = -3x + 6
y = 3x – 6
Step 3: y = 3x – 6
y = 3(2) – 6
y = 0
(2,0)
Algebra II
13
2. x – 3y = 4
6x – 2y = 4
Step 2: 6x – 2y = 4
6(3y + 4) – 2y = 4
18y + 24 – 2y = 4 16y = -20
y = -5/4 Step 1: x – 3y = 4
x = 3y +4
Step 3: (1/4, -5/4)
x = 3y + 4
x = 3(-5/4) + 4
x = 1/4
Algebra II
14
3. y = 2x – 5
8x – 4y = 20
Step 2: 8x – 4y = 20
8x – 4(2x – 5) = 20 8x – 8x + 20 = 20 20 = 20
0=0
Step 1: Y = 2x – 5 (already done)
True Statement!
Infinitely Many Solutions
Algebra II
15
4. -4x + y = 6
-5x – y = 21
Step 2: -5x – y = 21
-5x – (4x + 6) = 21
-5x – 4x – 6 = 21
-9x = 27
x = -3
Step 1: -4x + y = 6
y = 4x + 6
Step 3: (-3,-6)
y = 4x + 6
y = 4(-3) + 6
y = -6
Algebra II
16
¡ 
Steps for elimination:
1.  Make one of the variables have opposite
coefficients (multiply by a constant if necessary)
2.  Add the equations together and solve for the
remaining variable
3.  Substitute the value from step 3 into one of the
original equations to complete the ordered pair
Algebra II
17
2 30x – 15y = -15
12x + 15y = -27
6x – 3y = –3
42x + 0 = -42
4x + 5y = –9
42x = -42
42
42
1 5(6x – 3y = –3)
3(4x + 5y = –9)
x = -1
Solve the following
system by elimination Algebra II
18
Use x = -1 to find y
3
2nd equation:
4x + 5y = -9
4(-1) + 5y = -9
-4 + 5y = -9
+4
+4
5y = -5
5
5
y = -1
Algebra II
(-1, -1) 19
Solve the following
system by elimination 3x – y = 4
6x – 2y = 4
1
Algebra II
-2(3x – y = 4)
(6x – 2y = 4)
2
-6x + 2y = -8
6x – 2y = 4
0 + 0 = -4
0 ≠
= -4
False!
No Solution
20
Solve the following
system by elimination
2
6x + 10y = -12
3x + 5y = -6
-6x + 6y = 24
2x – 2y = -8
0 + 16y = 12
2(3x + 5y = -6)
16y = 12
1
-3(2x – 2y = -8)
16
16
y = 3/4
Algebra II
21
3
Use y = 3/4 to find x
1st equation:
3x + 5y = -6
3x+ 5(3/4) = -6
3x + 15/4 = -6
-15/4 -15/4
3x = -39/4
3
3
y = -13/4
Algebra II
(-13/4, 3/4) 22
Solve the following
system by elimination
-2x + y = -5
8x – 4y = 20
1
Algebra II
4(-2x + y = -5)
(8x – 4y = 20)
2 -8x + 4y = -20
8x – 4y = 20
0 + 0 = 0
= 0
0 =
True!
Infinitely Many
Solutions
23
1. 4x – 3y = 10
2x + 2y = 7
4. 3x + 2y = 8
2y + 4x = -2
5. 2x + 7y = 10
x + 4y = 9
2. Y = 3x – 5
2x + 3y = 8
3. X – 3y = 10
4x + 3y = 21
6. x – 3y = -6
x = 2y
Algebra II
24
1. 4x – 3y = 10
8x – 6y = 5
2. 3x + 3y = 10
2x – 2y = 15
M = 4/3, b= -10/3
M = -1, b = 10/3
M = 4/3 b = -5/6
M = 1, b = -15/2 No solution
Algebra II
One solution
25
3.
Algebra II
y = 2x + 8
2x – y = -8
4.
1/2x + 3y = 6
1/3x – 5y = -3
M = 2, b= 8
M = -1/6, b = 2
M = 2, b = 8
M = 1/15, b = 3/5 Infinitely
many One
solution
26
1. Your family is planning a 7 day trip to Florida.
You estimate that it will cost $275 per day in
Tampa and $400 per day in Orlando. Your total
budget for the 7 day is $2300. How many days
should you spend in each location? ¡  X = # of days in Tampa
¡  Y = # of days in Orlando
¡ 
¡ 
X + y = 7
275x + 400 y = 2300
Algebra II
27
2. You plan to work 200 hours this summer
mowing lawns or babysitting. You need to make
a total of $1300. Babysitting pays $6 per hour
and lawn mowing pays $8 per hour. How many
hours should you work at each job? ¡  X = # of hours babysitting
¡  Y = # of hours of mowing
¡ 
¡ 
X + y = 200
6x + 8y = 1300
Algebra II
28
3. You make small wreaths and large wreaths to
sell at a craft fair. Small wreaths sell for $8 and
large wreaths sell for $12. You think you can sell 40
wreaths all together and want to make $400. How
many of each type of wreath should you bring to the
fair?
¡  X = # small wreaths
¡  Y = # large wreaths
¡ 
¡ 
X + y = 40
8x
+ 12y = 400
Algebra II
29
4. You are buying lotions or soaps for 12 of
your friends. You spent $100. Soaps cost $5
a piece and lotions are $8. How many of
each did you buy? x = # of soaps
y = # of lotions
x + y = 12
5x + 8y = 100
Algebra II
30
5. Becky has 52 coins in nickels and dimes.
She has a total of $4.65. How many of each
coin does she have?
x = # of nickels
y = # of dimes
x + y = 52
.05x + .10y = 4.65
Algebra II
31
6. There were twice as many students as
adults at the ball game. There were 2500
people at the game. How many students and
how many parents were at the game? x = # of students
y = # of parents
x = 2y
x + y = 2500
Algebra II
32
1. Using substitution, solve the system:
{
3x + 4y = -4
x + 2y = 2
(-8, 5)
2. Using elimination, solve the system:
Algebra II
{
-3x + y = 11
5x – 2y = -16
(-6, -7)
33