solving systems of linear equations graphically 9.1 1.determine whether an ordered pair is a...
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Solving Systems of Linear Equations Graphically9.19.1
1. Determine whether an ordered pair is a solution for a system of equations.
2. Solve a system of linear equations graphically.3. Classify systems of linear equations in two unknowns.
System of equations: A group of two or more equations.
Solution for a system of equations: An ordered pair that makes all equations in the system true.
5 (Equation 1)
3 4 8 (Equation 2)
x y
x y
To Check a Solution to a System of Equations
1. Replace each variable in each equation with its corresponding value.
2. Verify that each equation is true.
Determine whether the ordered pair (3, 4) is a solution to the system of equations.
7 (Equation 1)
3 2 (Equation 2)
x y
y x
y = 3x – 2 4 = 3(3) – 2
4 = 7False
Because (3, 4) does not satisfy both equations, it is not a solution to the system of equations.
x + y = 73 + 4 = 7
7 = 7True
Determine whether the ordered pair (3, 2) is a solution to the system of equations.
x + y = 7 y = 3x – 2 3 + 2 = 7 2 = 3(3) – 2 1 = 7 2 = 11 False False
7 (Equation 1)
3 2 (Equation 2)
x y
y x
Because (3, 2) does not satisfy both equations, it is not a solution for the system.
Slide 5- 6Copyright © 2011 Pearson Education, Inc.
Which set of points is a solution to the system?
a) (–1, 1)
b) (–1, –1)
c) (0, 2)
d) (–3, 7)
2 4 2
3 2 1
x y
x y
9.1
Slide 5- 7Copyright © 2011 Pearson Education, Inc.
Which set of points is a solution to the system?
a) (–1, 1)
b) (–1, –1)
c) (0, 2)
d) (–3, 7)
2 4 2
3 2 1
x y
x y
9.1
A system of two linear equations in two variables can have one solution, no solution, or an infinite number of solutions.
2
5
4
x
y x
y
The graphs intersect at a single point. There is one solution.
3 2
3 1x
y x
y
2 8
2 4
4x y
x y
The equations have the same slope, the graphs are parallel. There is no solution.
The graphs are identical. There are an infinite number of solutions. (Not the same as all real numbers.)
Solving Systems of Equations Graphically
1. Graph each equation.
a. If the lines intersect at a single point, then the coordinates of that point form the solution.
b. If the lines are parallel, there is no solution.
c. If the lines are identical, there are an infinite number of solutions. They are the coordinates of all the points on that line.
2. Check your solution.
Solve the system of equations graphically.
Graph each equation:
y = 2 – x y = -x + 2 (0,2) m = -1
2x + 4y =12(0,3) (6,0) m = -½
Intersection:
y = 2 – x
2x + 4y = 12(2, 4)
2 (Equation 1)
2 4 12 (Equation 2)
y x
x y
The solution is (-2,4).
Solve the system of equations graphically.
(Equation 1)
(Equation 2)3 4
33
44x
y x
y
The slopes are the same, so the lines are parallel. The system has no solution
143
434
xy
xy
Solve the system of equations graphically.
2
(Equation 1
1
)
(Equation
4 2
)
2
2y x
x y
The equations are identical. All ordered pairs along the line are solutions.
12242
xyxy
Slide 5- 13Copyright © 2011 Pearson Education, Inc.
Solve the system of equations graphically.
a) (2, -3)
b) (2, -2)
c) No Solution
d) Infinite number of solutions
9.1
224
yx
xy
Slide 5- 14Copyright © 2011 Pearson Education, Inc.
Solve the system of equations graphically.
a) (2, -3)
b) (2, -2)
c) No Solution
d) Infinite number of solutions
9.1
224
yx
xy
Consistent system of equations: A system of equations that has at least one solution.
Inconsistent system of equations: A system of equations that has no solution.
Copyright © 2011 Pearson Education, Inc.
Classifying Systems of Equations
Consistent system with independent equations:
The system has a single solution at the point of intersection.
The graphs are different.
They have different slopes.
Consistent system with dependent equations:
The system has an infinite number of solutions.
The graphs are identical.
They have the same slope and same y-intercept.
Inconsistent system:
The system has no solution.
The graphs are parallel lines.
They have the same slope, but different y-intercepts.
Classify this system of equations.
a) Consistent with independent equations
b) Consistent with dependent equations
c) Inconsistent
9.1
12422
yx
xy
y = 2 – x
2x + 4y = 12(2, 4)
Classify this system of equations.
a) Consistent with independent equations
b) Consistent with dependent equations
c) Inconsistent
9.1
12422
yx
xy
y = 2 – x
2x + 4y = 12(2, 4)
Classify this system of equations.
a) Consistent with independent equations
b) Consistent with dependent equations
c) Inconsistent
9.1
443
343
yx
xy
Classify this system of equations.
a) Consistent with independent equations
b) Consistent with dependent equations
c) Inconsistent
9.1
443
343
yx
xy
Classify this system of equations.
a) Consistent with independent equations
b) Consistent with dependent equations
c) Inconsistent
9.1
12224
xyyx
Classify this system of equations.
a) Consistent with independent equations
b) Consistent with dependent equations
c) Inconsistent
9.1
12224
xyyx