Systems of Linear Equations

1
Systems of Linear Equations
2
Preliminaries

• How many solutions does the linear equation y
  2x - 1 have? Why?
• How many solutions does the linear equation y x
  3 have? Why?
• How many solutions do y 2x 1 and y x 3
  have in common?
• Can two linear equations have exactly two
  solutions in common? Explain.

3
Definitions

• Two or more linear equations form a system of
  linear equations.
• A solution of a system of linear equations is
  any ordered pair that makes all of the equations
  in the system true.

4
Example 1

• Determine if (2, 3) is a solution to the system
  of linear equations

y x 1 y 4x 5
5
Solution
Test point (2, 3) System of Linear Equation y
x 1 y 4x 5

• We verify if the given ordered pair makes all of
  the (two) equations in the system true
• Equation 1 3 2 1 ?
• Equation 2 3 (4)(2) 5 ?
• Because it satisfies all of the equations, the
  ordered pair (2, 3) is a solution to the given
  system of linear equation.

6
Example 2

• Determine if (3, -1) is a solution to the system
  of linear equations

2x 3y 3 x y 2
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Solution
Test point (3, -1) System of Linear
Equation 2x 3y 3 x y 2

• We verify if this satisfies the first equation
  (2)(3) (3)(-1) 3 ?
• Next we check if this satisfies the second
  equation (3) (-1) 2 ?
• Because it does not satisfy at least one
  equation, it is NOT a solution of the system of
  linear equations.

8
Exercises

• Determine if the given ordered pair is a solution
  to the given system of linear equations.
• a) (-1, 4)
• b) (3, -3)

y 3x 8 y 2x 6
No, because it does not satisfy the first
equation.

4x 2y 6 2x y 9
Yes, because it satisfies both of the equations.
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How do we find a solution?

• Using graphs
• Using tables
• Using algebraic properties

10
Solving by Graphs
Solution to system of equations

• Step 1
• Draw the graph
• of each equation.

• Step 2
• Get the point of intersection.

REVIEW GRAPHING LINEAR EQUATIONS
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Example 3
yx1
y3x 1

• Find the solution of the system of linear
  equations given below

y x 1 y 3x 1

• Solution (1, 2)
• Verify
• 2 1 1
• 2 (3)(1) - 1

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Exercise 2

• Solve the following system of linear equations
  using graphs.

y 3x 4 y 2x 2

x y -4 4x 2y 2
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Points for Discussion

• About two lines
• Can intersect in exactly one point
• Can be parallel
• Can coincide

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Other Points

• System of 3 linear equations in 2 variables
• System of n linear equations in 2 variables

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Points for Discussion

• Some problems with solving by graphs
• Requires very accurate graphs
• Needs verifying the solution
• Difficult to use in systems where the solution
  does not involve integers

16
Using algebraic solution

• Goal reduce the equations into a single
  equation involving only one variable.

17
End of Section
18
Appendix
19
Graphing Linear Equations

• A Quick Review

20
Recall

• Any equation that can be expressed as
• Ax By C (where A, B and C are fixed
  constants)
• has as a graph a line in the Cartesian Plane.
• Two points determine a line.

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How to Draw the graph of a Linear Equation

• Step 1 Determine 2 points that satisfy the
  equation.
• Step 2 Plot the two points.
• Step 3 Draw the line through the two points.

22
Example 1

• Sketch the graph of the linear equation
• 3x 2y 12

• Step 1 Determine 2 points
• x 0 -gt y 6
• (0, 6) is on the graph
• y 0 -gt x 4
• (4, 0 is on the graph)

• Step 2 Plot the 2 points

• Step 3 Draw a line passing through the 2 points

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Example 2

• Sketch the graph of the linear equation
• y 2x

• Step 1 Determine 2 points
• x 0 -gt y 0
• (0, 0) is on the graph
• x 3 -gt y 6
• (3, 6 is on the graph)

• Step 2 Plot the 2 points

• Step 3 Draw a line passing through the 2 points

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End of Review on Graphing Linear Equations
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