Ws-zerosolutions

Determining when equations have zero, ∞ solutions – 2009 By the end of this activity you will know how to determine when an equation or system of equations has no solution, one solution, or an infinite (inf ty) number of solutions. 1. An equation that results in a statement that is always true (such as 5 = 5) has an infinite number of solutions. An equation that results in a statement that is never true (3 = 4) has zero solutions, and equations that yields an open sentence, like x = 4, have one solution. Tell the number of solutions to each equation below. (a) 3x + 2(3x + 3) = 12x − 2(1.5x + 4)

(b)

3x−2 4

= .75x − .5

(c) 5x + 4 = 3x − 2

(d) 3x + 4x = 3x + 4x + 2 2. Systems of equations work the same way. The system

4x + 2y = 10 x + .5y = 3 has no solutions. This can be shown by solving the system until we reach a false statement.

x = 3 − .5y

Subtraction prop

(1)

4(3 − .5y) + 2y = 10 12 − 2y + 2y = 10

Substitution Distribution

(2) (3)

12 = 10

FALSE

(4)

Because 12 6= 10 there are no solutions to the system of equations. (a) Show that there are no solutions to the system 8x + 4y = 5 32 − 8y x= 16 3. Show that there are an infinite number of solutions x + 2y = 8 −1 x+4 y= 2 4. Write an equation with (a) zero solutions

(b) an infinite number of solutions

(c) only one solution

Solutions 1. Tell the number of solutions to each equation below. (a) 3x + 2(3x + 3) = 12x − 2(1.5x + 4) 3x + 6x + 6 = 12x − 3x − 8

(5)

9x + 6 = 9x − 8 6 6= −8

(6) (7)

No solutions

(b)

3x−2 4

= .75x − .5 3x − 2 = 3x − 2

(8)

−2 = −2

(9)

There is an infinite number of solutions

2. (a) Show that there are no solutions to the system 8x + 4y = 5 32 − 8y x= 16   32 − 8y + 4y = 5 8 16 8(32 − 8y) + 4y = 5 16 32 − 8y + 4y = 5 2 32 − 8y + 8y = 10 32 6= 10 So, there are no solutions to the system. 3. Show that there are an infinite number of solutions

(c) 5x + 4 = 3x − 2 5x = 3x − 6 2x = −6 x = −3

Substitution

(10) (11) (12)

One solution



x + 2y = 8 −1 x+4 y= 2 

−1 x+4 =8 2 2 x+− x+8=8 2 8=8

x+2

So, there are an infinite number of solutions (d) 3x + 4x = 3x + 4x + 2 7x = 7x + 2 0=2 No solutions

4. Write an equation with (13) (14)

(a) zero solutions: 3x − 4 = 3(x − 4) (b) an infinite number of solutions: 4x = 3x−(−x) (c) only one solution: 4x + 5 = 5x + 6