algebra 1b - palisades school district / overvie · · 2015-10-21algebra 1b chapter 7: ... it is...

Algebra 1B

Chapter 7: Expressions and Equations Containing Two Variables

Fall 2015

Name: ______________________________________ Period: ___________

Algebra 1B Chapter 7 Notes Fall 2015

Solving Systems Using Tables & Graphs

System of linear equations: Two or more equations with the same

variables solved _________________________.

Solution of a System of Linear Equations: Any ordered pair that ______________________ all equations in a system of equations. When

graphed, it is the point at which the graphs of each equation in a system of equations _____________________.

Determining if a point is a solution to a system of linear equations

a) Is (6, -1) a solution to the following system of equations?

𝑦 =1

3𝑥 − 3

𝑦 = −1

3𝑥 + 1

b) Is (2, -2) a solution to the following system of equations?

𝑦 = 3𝑥 + 4 −12𝑥 + 6𝑦 = −1

Determine the solution to a system of linear equations from a TABLE Complete the table for each equation. Then determine the solution.

𝑦 = 2𝑥 𝑦 = −1

2𝑥 + 5

X Y

-2

-1

0

1

2

Solution: _____________

X Y

-2

-1

0

1

2


Solving A System of Equations By Graphing

a) 𝑦 = 2𝑥 − 3

𝑦 = 𝑥 − 1

Your goal is to:

b) 𝑦 = 4𝑥 − 1 c) 𝑦 = −3

2𝑥 − 3

𝑦 = −𝑥 + 4 𝑥 + 2𝑦 = 2

Solution: _____________ Solution: _____________

Hint: Your goal is to…


Practice Problems:

1) 𝑥 + 𝑦 = 4 2) 𝑦 = 𝑥 + 2

𝑦 = −1 𝑦 = −2𝑥 + 2


3) 𝑦 = −2𝑥 + 1 4) 𝑦 = −1

4𝑥 + 2𝑦 = −6 𝑦 = 𝑥



5) −12𝑥 + 4𝑦 = 16 6) −2𝑥 + 𝑦 = 6

𝑦 − 4 = 3𝑥 8𝑥 − 4𝑦 = 2


7) Is (-3, -1) a solution to the following system of equations?

𝑦 =4

3𝑥 + 3

𝑦 = −2

3𝑥 − 3

Let’s Summarize:

Lines with different slopes and different y-intercepts…

Lines with the same slope and different y-intercepts…

Lines with the same slope and same y-intercept…

http://www.ixl.com/math/algebra-1/solve-a-system-of-equations-by-graphing

http://www.ixl.com/math/algebra-1/solve-a-system-of-equations-by-graphing


3-2 NOTES (Part 1): Solving Systems of Equations by Substitution

Is there an easier way to find the EXACT solution, without graphing??

𝑦 = −4𝑥 − 7

𝑥 − 𝑦 = 2

Let’s try a few more..

a) 𝑦 = 2𝑥 b) 𝑦 = 𝑥 + 1

7𝑥 − 𝑦 = 15 𝑦 = 2𝑥 − 1

c) 𝑦 = −1

2𝑥 + 4 d) 𝑥 − 3𝑦 = 14

2𝑥 + 2𝑦 = 2 𝑥 − 2 = 0

Yes!! Use substitution…

HINT: Your goal is to…


What about ‘No solution’?

What kind of lines have no solution??

𝑥 + 2𝑦 = −16 2𝑥 + 4𝑦 = 2

What about ‘Infinitely Many Solutions’?

What kind of lines have infinitely many solutions??

4𝑥 + 4𝑦 = −4 𝑦 = −𝑥 − 1


1) 2𝑥 + 𝑦 = 0 2) 𝑦 = −𝑥 − 5 10𝑥 + 5𝑦 = 8 3𝑥 + 3𝑦 = −15


PRACTICE PROBLEMS

Solving Systems of Equations - Substitution

Solve each system of equations using the substitution method.

1) 4𝑥 + 12𝑦 = 4 2) 𝑥 + 4𝑦 = 8

𝑦 = 5𝑥 + 11 3𝑥 = 3 − 12𝑦

3) 𝑦 = −3𝑥 − 1 4) 12𝑥 − 2𝑦 = 0

2𝑥 − 3𝑦 = −8 −7𝑥 + 𝑦 = −1


5) −2𝑥 + 4𝑦 = 22 6) 𝑦 =1

7𝑥 + 3

𝑥 = 2𝑦 − 11 𝑥 − 7𝑦 = −4

7) 𝑥 − 3𝑦 = −4 8) 3𝑥 − 𝑦 = 7

2𝑥 + 3𝑦 = −5 −14 + 6𝑥 = 2𝑦

9) 2𝑥 − 6𝑦 = −2 10) 𝑥 = 10𝑦 + 34

𝑥 = 3𝑦 − 1 −7𝑥 + 𝑦 = −31


Why have more than one method??

Solve the following systems of linear equations by graphing and then by substitution. Compare your solutions. What do you notice?

1.) 23 xy

43 xy

Graphing: Substitution:

Solution: ____________________ Solution: ___________________

2.) 24 xy

3 yx


Solution: ____________________ Solution: _____________________


3.) 2 yx

4 xy


Solution: ____________________ Solution: ___________________

4.) 1024 yx

33 yx


Solution: ____________________ Solution: _____________________


3-2 NOTES (Part 2): Solving Systems of Equations by Elimination

*Sometimes you are given a system of equations that contain two equations that are difficult to isolate one variable. There is an easier

method than substitution to use to solve the system!!

It is called “__________________________”

3𝑥 + 5𝑦 = 15

−3𝑥 + 6𝑦 = 7


a) 5𝑥 − 6𝑦 = −32 b) 6𝑥 − 3𝑦 = 3 3𝑥 + 6𝑦 = 48 −6𝑥 + 5𝑦 = 3

What happens when the system doesn’t contain terms that are

opposites??

2𝑥 + 5𝑦 = 12 𝑥 − 3𝑦 = −2

HINT: Make opposites by multiplying!!

HINT: Your goal is to…



a) −2𝑥 + 15𝑦 = −32 b) 2𝑥 + 𝑦 = −1 7𝑥 − 5𝑦 = 17 3𝑥 + 𝑦 = 0

c) 𝑥 − 6𝑦 = −10 d) 𝑥 − 𝑦 = 3

3𝑥 + 2𝑦 = 0 𝑥 + 4𝑦 = −7


Elimination Practice Problems Solve each system of equations using the elimination method. Show ALL work! Be sure you write your answer as (x,y)!

1) 2)

3) 4)

5) 6)


Compare & Contrast Solving System of Equations by Graphing, Substitution, Elimination Describe, in words, HOW to solve using each method:

- Graphing: ____________________________________________________________

________________________________________________________________________

________________________________________________________________________________________________________________________________________________

- Substitution: __________________________________________________________ ________________________________________________________________________

________________________________________________________________________________________________________________________________________________

- Elimination: ___________________________________________________________ ________________________________________________________________________

________________________________________________________________________________________________________________________________________________

Decide which method you would use to solve the given system of equations. Then, explain why you would choose a certain method. (You do not need to solve.)

1) 2𝑥 − 𝑦 = −3 2) 𝑦 = 2𝑥

𝑥 + 𝑦 = 9 3𝑥 + 𝑦 = 10

3) 𝑦 = 𝑥 − 5 4) 8𝑥 − 3𝑦 = 15

𝑦 = −2

3𝑥 + 6 7𝑥 − 4𝑦 = 20


Solve each system of equations by the method of your choice. (You only need to solve using one method.)

5) 4𝑥 + 3𝑦 = 12 2𝑥 − 5𝑦 = −20

GRAPHING: SUBSTITUTION/ELIMINATION:

Solution: _____________

6) 𝑦 = 4𝑥 − 7

𝑦 = 2𝑥 + 9 GRAPHING: SUBSTITUTION/ELIMINATION:

Solution: _____________


Solving Systems of Inequalities How did we find the solution to a system of EQUATIONS? What do you think we will do to find the solution to a system of INEQUALITIES? Solution Set of a System of Inequalities:

Graph the following systems of inequalities. Label the solution set ‘S.’

c) 𝑦 ≥ 𝑥 + 5 b) 3𝑥 − 𝑦 ≥ 2 𝑦 ≤ −2𝑥 + 1 𝑦 ≥ −5


c) 𝑥 > −1 d) 𝑦 ≤2

3𝑥 + 3

𝑦 ≥1

5𝑥 + 2 𝑦 > 2𝑥 − 2

e) 𝑥 > −1 f) 𝑦 ≤ −7 𝑦 ≥ 2 𝑦 ≥ −2

Summarize for yourself…

algebra 1b - palisades school district / overvie · · 2015-10-21algebra 1b chapter 7: ... it is...

Documents