3.1 solving linear systems by graphing 10/1/12. solution of a system of 2 linear equations: is an...
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3.1 Solving Linear Systems by Graphing
10/1/12
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Solution of a system of 2 linear
equations:
Is an ordered pair (x, y) that satisfies both equations.Graphically, it’s the point where the lines intersect.
VocabularySystem of 2
Linear Equations:
A system consisting of two linear equations in two variables.Ex: 6x – 2y = 8 3x – y = 4
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Tell whether the ordered pair (3, 4) is a solution of
-2x + y = -2 4x – 2y = 3Substitute 3 for x and 4 for y in BOTH equations. -2(3) + 4 = -2 - 6 + 4 = -2
4(3) – 2(4) = 3 12 – 8 = 3
Answer: Not a Solution
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Tell whether the ordered pair (3, 4) is a solution of x + 2y = 11 2x – y = 2
Substitute 3 for x and 4 for y in BOTH equations. 3 + 2(4) = 11 3 + 8 = 11
2(3) – 4 = 2 6 – 4 = 2
Answer: Solution
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Solve the system by graphing. Then check your solution.
3y – x= +
9y 2x= +
Solve a System by GraphingExample 1
ANSWER ( )2, 5–
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y = - x + 3 5= -(-2) + 3 5= 5
y = 2 x + 9 5 = 2(-2) + 95 = -4 + 95 = 5
You can check the solution by substituting -2 for x and 5 for y into the original equations.
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Example 2
ANSWER ( )2, 3
Solve a System by Graphing
Solve the system by graphing. Then check your solution algebraically.
33x – y =
8x + 2y =
In slope int. form: y = 3x - 3
In slope int. form: y = - x + 4
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Example 2 Solve a System by Graphing
You can check the solution by substituting 2 for x and 3 for y into the original equations.
Equation 1 Equation 2
33x – y = 8x + 2y =
( )2 33 – 3 =? 2 8+ =?( )32
36 – 3 =? 2 8+ =
?6
33 = 88 =
ANSWER ( ).2, 3The solution of the system is
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Extra Example
1. x 3y =– 1
x y =+ 1– –
ANSWER ( )1, 0
Solve the system by graphing. Then check your solution.
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Checkpoint
2. =
2x 3y =– 6
ANSWER ( )6, 2
Solve a System by Graphing
Solve the system by graphing. Then check your solution.
2+ 4yx–
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Homework WS 3.1
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Number of Solutions
1 solution: the lines have different slopes
Infinitely many solutions:the lines have the same equation.
No solution:the lines are parallel (same slope)
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Systems with Many or No SolutionsExample 3
Tell how many solutions the linear system has.
a. 1=y–2x
+ =2y–4x 2–
b.
+ =2yx 1
+ =2yx 4
12
2 2 2
242
4 4
224
xy
xy
x x
yx
12
1- 1- 1-
12
2 2
12
xy
xy
x -x
yx
22
1
2 2 2
42
42
xy
xy
-x x
yx
2
1
2
1
2 2 2
12
12
xy
xy
-x x
yx
Infinitely many solutions:the lines have the same equation.
No solution:the lines are parallel (same slope)
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Tell how many solutions the linear system has.
Checkpoint
ANSWER 0
Write and Use Linear Systems
4. 5=4y–x
+ =4y–x 5–
3. + =3y2x 1
+ =6y4x 3
ANSWER infinitely many solutions
ANSWER 1
5. 5=5y–x
+ =5yx 5