**Semester 2 Honors Algebra 1 Study Guide: Chapter 6**

**By: Katie Schulz**

Hour 2

Hour 2

**Table of Contents**

**Chapter 6**

**Objectives:**

Solving linear systems by graphing

Solving linear systems using algebra

Soving Sysytems of linear inequalities

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 = 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!!