geometry section 0-8 11-12

Section 0-8Systems of Equations

Monday, September 19, 2011

Essential Question

How do you use graphing, substitution, and elimination to solve systems of linear equations?


Vocabulary

1. System of Equations:

2. Substitution:

3. Elimination:


Vocabulary

1. System of Equations: Two or more equations with the same two variables that you solve at the same time

2. Substitution:

3. Elimination:


Vocabulary


2. Substitution: Plugging in a number or expression for a variable

3. Elimination:


Vocabulary


2. Substitution: Plugging in a number or expression for a variable

3. Elimination: Using addition and multiplication to eliminate parts of a system to achieve a solution


Types of Solutions


Types of Solutions

One solution: a point


Types of Solutions


No solutions: parallel lines


Types of Solutions


No solutions: parallel lines

Infinitely many solutions on the line: same line


Example 1

Solve the system by graphing.

x

y


Example 1


x

y

y =2x −4y = −x +5

⎧⎨⎪

⎩⎪


Example 1


x

y

y =2x −4y = −x +5

⎧⎨⎪

⎩⎪

(3, 2)


Example 1


x

y

y =2x −4y = −x +5

⎧⎨⎪

⎩⎪

(3, 2)

Check:


Example 1


x

y

y =2x −4y = −x +5

⎧⎨⎪

⎩⎪

(3, 2)

Check:

2=2(3)−4


Example 1


x

y

y =2x −4y = −x +5

⎧⎨⎪

⎩⎪

(3, 2)

Check:

2=2(3)−4 2=6−4


Example 1


x

y

y =2x −4y = −x +5

⎧⎨⎪

⎩⎪

(3, 2)

Check:

2=2(3)−4 2=6−4

2= −3+5


Solve a System of Equations by Substitution



1. Solve one equation for one variable (your choice)




2. Substitute the expression from the equation into the other equation





3. Solve for the variable and substitute back into the original equation to find the other variable





3. Solve for the variable and substitute back into the original equation to find the other variable

4. Rewrite your answer as an ordered pair and check it!


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9

y =6


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9

y =6

Check:


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9

y =6

Check:

3+6=9


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9

y =6

Check:

3+6=9

10(3)+6=12(3)


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9

y =6

Check:

3+6=9

10(3)+6=12(3) 30+6=36


Example 2

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9

y =6

Check:

3+6=9

10(3)+6=12(3) 30+6=36

(3, 6)


Solve by Elimination



1. Choose a variable to eliminate (your choice).




2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations.





3. Solve for the remaining variable.






4. Plug back into an original equation to find the other variable.






4. Plug back into an original equation to find the other variable.

5. Check and rewrite the answer.


Example 3

Solve by combining the equations

7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −17


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16 +2 +2


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3

y = 6


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3

y = 6

Check:


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3

y = 6

Check:

7(−1)+ 2(6) = 5


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3

y = 6

Check:

7(−1)+ 2(6) = 5 −7 + 12 = 5


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3

y = 6

Check:

7(−1)+ 2(6) = 5 −7 + 12 = 5

2(−1)+ 3(6) = 16


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3

y = 6

Check:

7(−1)+ 2(6) = 5 −7 + 12 = 5

2(−1)+ 3(6) = 16

−2 + 18 = 16


Example 3


7x + 2 y = 52x + 3 y = 16

⎧⎨⎪

⎩⎪

( ) (3)( ) (−2)

21x + 6 y = 15 −4x − 6 y = −32

17x = −1717 17

x = −1

2(−1)+ 3 y = 16

−2 + 3 y = 16

3 y = 18 +2 +2

3 3

y = 6

Check:

7(−1)+ 2(6) = 5 −7 + 12 = 5

2(−1)+ 3(6) = 16

−2 + 18 = 16

(−1,6)


Example 4Solve each system of equations. Check your solution.

a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14

−14= −14



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14

−14= −14What’s going on here?



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14


−2 y = −x +7



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14


−2 y = −x +7

y =

12

x − 72



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14


−2 y = −x +7

y =

12

x − 72

4 y =2x −14



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14


−2 y = −x +7

y =

12

x − 72

4 y =2x −14

y =

12

x − 72



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14


−2 y = −x +7

y =

12

x − 72

4 y =2x −14

y =

12

x − 72

These are the same lines!



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14

−4 y −14+4 y = −14


−2 y = −x +7

y =

12

x − 72

4 y =2x −14

y =

12

x − 72

These are the same lines!

Infinitely many solutions on the line.



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3

−4x +4x −4=3



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3

−4x +4x −4=3 −4=3



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3

−4x +4x −4=3 −4=3

14 y = 4x +3



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3

−4x +4x −4=3 −4=3

14 y = 4x +3

y =

27

x + 314



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3

−4x +4x −4=3 −4=3

14 y = 4x +3

y =

27

x + 314

These lines are parallel.



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3

−4x +4x −4=3 −4=3

14 y = 4x +3

y =

27

x + 314

These lines are parallel.There are no solutions.


When solving a system you get:



One solution when:



One solution when:

No solutions when:



One solution when:

No solutions when:

An infinite number of solutions on the line when:


Problem Set


Problem Set

p. P18 #1-15 all

“I have failed many times, and that’s why I am a success.” - Michael JordanMonday, September 19, 2011

geometry section 0-8 11-12

Education