1. 5.4 Solving Special Systems of Linear Equations

2. Solutions of Systems of Linear Equations
A system of linear equations can have one solution, no solution or infinitely many solutions.

3. Example 1 Solving a System: No Solution
Solve the system of linear equations. y = 2x + 1 y = 2x – 5 Solve by substitution: 2x + 1 = 2x – 5 -2x -2x 1 = -5 Since that is not true, the system has no solution.

4. More on Example 1
Solve the system of linear equations. y = 2x + 1 y = 2x – 5 Solve by Graphing: Since the lines are parallel (same slope) , they will never intersect. So, the system has no solution.

5. Example 2 Solving a System: Infinitely Many Solutions
Solve the system of linear equations -2x + y = 3 -4x + 2y = 6 Solve by elimination: 4x - 2y = = 0 Since this is true the system has infinitely many solutions.
Multiply equation 1 by -2
4x - 2y = -6
-4x + 2y = 6

6. More on Example 2
Solve the system of linear equations. -2x + y = 3 -4x + 2y = 6 Solve by Graphing: Since the lines are the same line when graphed. They have infinitely many solutons.
x int: (-1.5, 0) y int: (0,3)
x int: (-1.5, 0) y int: (0,3)

7. Infinitely many solutions
You try!
Solve the system of linear equations. 1) 2)
x + y = 3
2x + 2y = 6
y = -x + 3
2x + 2y = 4
Infinitely many solutions
No solution

8. Example 3 Modeling with Mathematics
The perimeter of the trapezoidal piece of land is 48 kilometers. The perimeter of the rectangular piece of land is 144 kilometers. Write and solve a system of linear equations to find the values of x and y. Perimeter of trapezoid: 2x + 6y + 4x + 6y = 48 6x + 12y = 48 Perimeter of rectangle: 9x + 18y + 9x + 18y = x + 36y = 144

9. System of linear equations: 6x + 12y = 48 18x + 36y = 144 Solve by elimination: -18x – 36y = = 0
Multiply by -3
-18x – 36y = -144
18x + 36y = 144
Since this is true, this means that there are infinitely many solutions to x and y. However, x and y must be positive since we can not have a negative side length. So, there are infinitely many positive solutions for x and y.