5 4 Solving Special Systems of Linear Equations

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5. 4 Solving Special Systems of Linear Equations

5. 4 Solving Special Systems of Linear Equations

Solutions of Systems of Linear Equations A system of linear equations can have one

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

Example 1 Solving a System: No Solution Solve the system of linear equations. y

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

More on Example 1 Solve the system of linear equations. y = 2 x

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

Example 2 Solving a System: Infinitely Many Solutions Solve the system of linear equations

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

More on Example 2 Solve the system of linear equations. -2 x + y

More on Example 2 Solve the system of linear equations. -2 x + y = 3 x int: (-1. 5, 0) y int: (0, 3) -4 x + 2 y = 6 x int: (-1. 5, 0) y int: (0, 3) Solve by Graphing: Since the lines are the same line when graphed. They have infinitely many solutons.

You try! Solve the system of linear equations. 1) x + y = 3

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

Example 3 Modeling with Mathematics The perimeter of the trapezoidal piece of land is

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: 2 x + 6 y + 4 x + 6 y = 48 6 x + 12 y = 48 Perimeter of rectangle: 9 x + 18 y + 9 x + 18 y = 144 18 x + 36 y = 144

System of linear equations: Multiply by -3 6 x + 12 y = 48

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

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