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Semester 2 Honors Algebra 1 Study Guide: Chapter 6

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Katie Schulz

on 26 May 2014

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Transcript of Semester 2 Honors Algebra 1 Study Guide: Chapter 6

Semester 2 Honors Algebra 1 Study Guide: Chapter 6
By: Katie Schulz
Hour 2

Table of Contents
Chapter 6
Objectives:
Solving linear systems by graphing

Solving linear systems using algebra

Soving Sysytems of linear inequalities

6.1 Solving Linear Systems by Graphing
Vocabulary:

System of linear Equations:
Consists of two or more linear equations in the same variables.

Example:
x+2y=7
3x-2y=5

Solution of a system of linear equation:
In two variables is an ordered pair that satisfies each equation in the system.

Consistent independent system:
A linear system that has one solution.
Solving a linear system using the Graph-and-check method
Step 1:

(Write both equation in slope-intercpet form)
Graph both equations on the same coordinate plane.

Step 2:
Estimate the cordinates where the two graphs intersect
(point of intersection)

Step 3:
Check the coordinates by subsitituting the numbers into each equation of the orginal linear system
Examples for Graph-and-check
Example:
x+2y=7
3x-2y=5
Try it yourself
2y = 6x + 8
4x + y = -3
turn it into slope intercept form
2y = 6x + 8
(divide by 2 on both sides)
y = 3x + 4

4x + y = -3
(subtract 4x on both sides)
y = -3 - 4x
y = 3x + 4
y = -4x - 3
Try it the answer on the next slide
Answer (-1, 1
)
Answer!
No peaking
Check your answer
2y = 6x + 8
4x + y = -3
x = -1 y = 1
2(1) = 6(-1) + 8

2 = -6 + 8

2=2
4(-1) + (1) = -3

-4 + 1 = -3

-3 = -3
BOTH SIDES CHECKED OUT!
6.1 = Slide #6
6.2 = Slide #14
6.3 = Slide #22
6.4 = Slide #32
concept summary = Slide #37
6.5 = Slide #38
6.6 = Slide #48

6.2: Solve Linear Systems by Substitution
Vocabulary:

System of linear equation-
Consists of two or more linear equations in the same variables.

Solving a Linear System Using the Substitution Method
Step 1:

Solve one of the equations for one of its variables.
(solve for a variable that has a coefficient of 1 or -1)

Step 2:

Subsitute the expression from Step 1 into the other equation and solve for the other variable

Step 3:

Subsitute the value from Step 2 into the revised equation from Step 1 and solve.
Example for Substitution Method
Example:
y = 3x + 2
x + 2y = 11
Try it yourself!!
5x - 4y = -1
y = 6x + 5
Normally you would solve for a variable, but "y" is already solved

Plug in the y equation into the other equation
5x - 4(6x + 5) = -1

5x - 24x - 20 = -1

-19x -20 = -1
+20 +20
-19x = 19
(divide by -19 on both sides)

x= -1
Then plug in x into one of the orignal equations to find "y"
y = 6(-1) + 5

y = -6 + 5

y = -1
answer: (-1, -1)

Check the answer
5(-1) - 4(-1) = -1
-5 + 4 = -1
-1 = -1
With both equations
-1= 6(-1) + 5
-1 = -6 + 5
-1 =-1
Story Problem!
You own a skate rental. You rent out both figure and hockey skates. 35 people rented skates today, and you made a profit of $190. You rent Figure skates for $5 per hour and hockey skates for $8 per hour.
Find out how many people bought each type of skate.
The equations!!
f + h = 35
5f + 8h = 190
f = h + 35
Solve for a variable
5(h + 35) + 8h = 190

5h + 175 + 8h = 190

5h + 175 = 190
- 175 -175
5h = 25
(divide both sides by 5)
h = 5
plug it into the other equation
Plug h into the orginal equations to find f (next slide)
find f
f + h = 35

f + 5 = 35
- 5 - 5
f = 30
answer:
20
people rented figure skates
15
people rented hockey skates
Check by pluging in 20 and 15
f + h = 35

30 + 5= 35

35 = 35
5f + 8h = 190

5(30) + 8(5) = 190

150 + 40 = 190

190 = 190
6.3 Solve Linear Systems by Adding or Subtracting
Vocabulary:
system of linear equations-
Consists of two or more linear equations in the same variables.
Solving a Linear System using the Elimination Method
Step 1:
Get opposites

Step 2:
Add the equtions to eleminate one variable

Step 3:
Solve the resulting equation for the other variable

Step 4:
Substitute in either orginal equation to find the valure of the other variable
Example
2x + 3y = 11
-2x + 5y = 13
Try it yourself
4x - 5y = 5
5y = x + 10
Line up terms to match the variables
4x - 5y = 5
5y = x + 10
-x -x
-x + 5y = 10
-x + 5y = 10
No peaking answer next slide
4x - 5y = 5
-x + 5y = 10
add the terms together
3x = 15
divide by 3 on each side
x = 5
practice problem
Plug x into the original equation
4x - 5y = 5

4(5) -5y = 5

20 - 5y = 5
-20 -20
-5y = -15
y = 3
answer: (5, 3)
Check your answer
4x - 5y = 5
5y = x + 10
4(5) - 5(3) = 5

20 - 15 = 5

5 = 5
5(3) = 5 + 10

15 = 5 + 10

15 = 15
Both problems checked out!!
Other great websites to get practice
Links:
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/solving-systems-graphically

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/e/graphing_systems_of_equations

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/king-s-cupcakes---solving-systems-by-elimination

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/simple-elimination-practice

http://www.purplemath.com/modules/systlin1.htm

http://www.mathsisfun.com/algebra/systems-linear-equations.html
Story Problem!!
A kayaker travels 12 miles upstream (against the current) and 12 miles downstream (with the current). When the kayker traveled went upstream it took 3 hours, and when it when downstream it took 2 hours.
Find the speed of the kayaker in still water.
6.4 Solve Linear systems by Multiplying First
Least Common multiple-
the lowest common multiple that two or more more numbers have in common.
The equations
First find the rate they are going for both upstream and downstream
distance = (rate)(time)
upstream:

12 = 3r

4 = r
downstream:

12 = 2r

6 = r
Upstream: x - y = 4
Downstream: x + y = 6
Solve equations
x - y = 4
x + y = 6

2x = 10
(divide by 2 on both sides)

x = 5
plug x into the equation
5 - y = 4
-5 -5
-y = -1
(divide by negative 1 on both sides)
y = 1
Answer = (5, 1)
Average speed of the kayak in still water is 5 miles per hour. Speed of the current is 1 mile per hour.
Check answer
x - y = 4

5 -1 = 4

4 = 4
x + y = 6

5 + 1 = 6

6 = 6
example:
6x + 5y = 19
2x + 3y = 5
Try it yourself!
7x + 2y = 26
(multiply by 5)
10x -5y = -10
(multiply by 2)
35x + 10y = 130
20x - 10y = -20

55x = 110
x = 2
7x + 2y = 26
10x - 5y = -10
No peaking!! (the answer is one the next slide)
Solve the problem
plug the x into the orginal equation
7(2) + 2y = 26

14 + 2y = 26
-14 -14

2y = 12

y = 6
the answer: 2, 6
Check the answer
7x + 2y = 26

7(2) + 2(6) = 26

14 + 12 = 26

26 = 26
10(2) - 5(6) = -10

20 - 30 = -10

-10 = -10
Both answers checked out!!!!!
Let's Review!!! the concepts so far
6. 5 Solve Special Types of Linear Systems
Vocabulary:

Inconsistent system-
A linear system with no solutions

Consistent dependent system-
A linear system with infinately number of solutions
Number of Solutions of a Linear System
Example of Linear system with no solutions
Example of a linear system with infinitely many solutions
Try it yourself
2y = 8x + 4
-4x + y = 4
-4x + y = 4
4x 4x
y = 4 + 4x
2y = 8x + 4
y = 4x + 4
(multiply by -2)
2y = 8x + 8
-2y = -8x - 8
0 = 0
answer: infinte many solutions
2y = 8x + 4
y = 4x + 4
line the terms up
Solve the equation
Extension: use Picewise Function
Vocabulary:
Piecewise functions-

At least two equations, each of which applies to a different part of the function's dmain. An example is given below.

example:





Step Functions-
A piecewise function that is defined by a constant value over each part of its domain
Graphing a piecwise function
Step 1:
To the left of x = -1. Use a closed dot at (-1,0) because the equation applies when x = -1.

Step 2:
From x = -1 to x = 2, graph y = 3. Use open dots at (-1,3) and (2,3) because the equation does not apply when x = -1 or when x = 2.

Step 3:
To the right of x = 2, graph
y = 2x - 5. Use a closed dot at (2, -1) because the equation applies when
x = 2.
Write a piecewise function
Example of a step function
6.6 Solve Systems of Linear Inequalities
Vocabulary:
System of linear inequalities-
in two variables, or simply a system of inequalities, consists of two or more linear inequalities in the sam variables.

Solution of a system of linear inequalitites-
is an ordered pair that is a solution of each inequality in the system.

Graph of a system of linear inequalities-
is the graph of all solution of the system.
Key Concepts
Example of a graph a system of two linear inequalities
Example of graphing a system of three linear inequalities
Example of writing a system of linear inqualities
Story Problem!!
Full transcript