Question

Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and infinite solutions. • Put a circle around the two equations that have infinite solutions. • Put a square around the two equations that have no solution. • Underline the two equations that have one solution. 2x+3=2x+7 7x+3=7x+3 4x+22=8x+10 5x+5=5x+4 3x+18=5x+8 3(4x+8)=4(3x+6) What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.

Answer 1

2x+3=2x+7
This simplifies to
3 = 7
NO solutions

7x+3=7x+3
This works for any old value of x, infinite number of solutions

4x+22=8x+10
4 x = 12
x = 3
Whew,finally one solution :)

5x+5=5x+4
oh my, 5 = 4?
No solution again

3x+18=5x+8
2 x = 10
x = 5
one solid solution

3(4x+8)=4(3x+6)
12 x + 24 = 12 x + 24
any old x :)

Answer 2

Consider the two equations you circled, which form a system with infinite solutions.
Solve the equations.

Answer 3

Consider the two equations you put a square around, which form a system with no
solution. Solve the equations.

Answer 4

The two equations with infinite solutions are:

7x+3=7x+3
3(4x+8)=4(3x+6)

The two equations with no solution are:

4x+22=8x+10
5x+5=5x+4

The two equations with one solution are:

2x+3=2x+7
3x+18=5x+8

Equations that have one solution have different coefficients or constants for the variable being solved for. Equations with no solution have coefficients or constants such that the variable disappears, resulting in an inconsistency. Equations with infinite solutions have coefficients and constants that result in the same expression on both sides of the equation.