geometry section 0-8 11-12
Post on 30-Apr-2015
619 Views
Preview:
DESCRIPTION
TRANSCRIPT
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?
Monday, September 19, 2011
Vocabulary
1. System of Equations:
2. Substitution:
3. Elimination:
Monday, September 19, 2011
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:
Monday, September 19, 2011
Vocabulary
1. System of Equations: Two or more equations with the same two variables that you solve at the same time
2. Substitution: Plugging in a number or expression for a variable
3. Elimination:
Monday, September 19, 2011
Vocabulary
1. System of Equations: Two or more equations with the same two variables that you solve at the same time
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
Monday, September 19, 2011
Types of Solutions
Monday, September 19, 2011
Types of Solutions
One solution: a point
Monday, September 19, 2011
Types of Solutions
One solution: a point
No solutions: parallel lines
Monday, September 19, 2011
Types of Solutions
One solution: a point
No solutions: parallel lines
Infinitely many solutions on the line: same line
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
(3, 2)
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
(3, 2)
Check:
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
(3, 2)
Check:
2=2(3)−4
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
(3, 2)
Check:
2=2(3)−4 2=6−4
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
(3, 2)
Check:
2=2(3)−4 2=6−4
2= −3+5
Monday, September 19, 2011
Example 1
Solve the system by graphing.
x
y
y =2x −4y = −x +5
⎧⎨⎪
⎩⎪
(3, 2)
Check:
2=2(3)−4 2=6−4
2= −3+5
Monday, September 19, 2011
Solve a System of Equations by Substitution
Monday, September 19, 2011
Solve a System of Equations by Substitution
1. Solve one equation for one variable (your choice)
Monday, September 19, 2011
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
Monday, September 19, 2011
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
Monday, September 19, 2011
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
4. Rewrite your answer as an ordered pair and check it!
Monday, September 19, 2011
Example 2
x + y =910x + y =12x⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 2
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
Monday, September 19, 2011
Example 2
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x
Monday, September 19, 2011
Example 2
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
Monday, September 19, 2011
Example 2
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x
Monday, September 19, 2011
Example 2
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
Monday, September 19, 2011
Example 2
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
Monday, September 19, 2011
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
Monday, September 19, 2011
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:
Monday, September 19, 2011
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
Monday, September 19, 2011
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)
Monday, September 19, 2011
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
Monday, September 19, 2011
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)
Monday, September 19, 2011
Solve by Elimination
Monday, September 19, 2011
Solve by Elimination
1. Choose a variable to eliminate (your choice).
Monday, September 19, 2011
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.
Monday, September 19, 2011
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.
Monday, September 19, 2011
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.
Monday, September 19, 2011
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.
5. Check and rewrite the answer.
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)( ) (−2)
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)( ) (−2)
21x + 6 y = 15
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)( ) (−2)
21x + 6 y = 15 −4x − 6 y = −32
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)( ) (−2)
21x + 6 y = 15 −4x − 6 y = −32
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)( ) (−2)
21x + 6 y = 15 −4x − 6 y = −32
17x = −17
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)( ) (−2)
21x + 6 y = 15 −4x − 6 y = −32
17x = −1717 17
Monday, September 19, 2011
Example 3
Solve by combining the equations
7x + 2 y = 52x + 3 y = 16
⎧⎨⎪
⎩⎪
( ) (3)( ) (−2)
21x + 6 y = 15 −4x − 6 y = −32
17x = −1717 17
x = −1
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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:
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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
Monday, September 19, 2011
Example 3
Solve by combining the equations
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)
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
−2 y = −x +7
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
−2 y = −x +7
y =
12
x − 72
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
y =
12
x − 72
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
y =
12
x − 72
These are the same lines!
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
−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.
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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?
−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.
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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.
Monday, September 19, 2011
Example 4Solve each system of equations. Check your solution.
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.
Monday, September 19, 2011
When solving a system you get:
Monday, September 19, 2011
When solving a system you get:
One solution when:
Monday, September 19, 2011
When solving a system you get:
One solution when:
No solutions when:
Monday, September 19, 2011
When solving a system you get:
One solution when:
No solutions when:
An infinite number of solutions on the line when:
Monday, September 19, 2011
Problem Set
Monday, September 19, 2011
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
top related