Question 1: Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and infinite solutions.

Question

• 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)

Answer 1

To determine the number of solutions for each equation, we need to simplify them:

2x + 3 = 2x + 7
Here, we can subtract 2x from both sides:
3 = 7
This equation is contradictory, so it has no solution.

7x + 3 = 7x + 3
This equation is true for all values of x. It has infinite solutions.

4x + 22 = 8x + 10
Subtracting 4x from both sides, we get:
22 = 4x + 10
Subtracting 10 from both sides, we obtain:
12 = 4x
Dividing both sides by 4, we have:
3 = x
This equation has one solution.

5x + 5 = 5x + 4
Subtracting 5x from both sides, we get:
5 = 4
This equation is contradictory, so it has no solution.

3x + 18 = 5x + 8
Subtracting 3x from both sides, we obtain:
18 = 2x + 8
Subtracting 8 from both sides, we have:
10 = 2x
Dividing both sides by 2, we get:
5 = x
This equation has one solution.

3(4x + 8) = 4(3x + 6)
Distributing on both sides, we obtain:
12x + 24 = 12x + 24
This equation is true for all values of x. It has infinite solutions.

Therefore, the equations with one solution are:
- 4x + 22 = 8x + 10
- 3x + 18 = 5x + 8

The equations with no solution are:
- 2x + 3 = 2x + 7
- 5x + 5 = 5x + 4

The equations with infinite solutions are:
- 7x + 3 = 7x + 3
- 3(4x + 8) = 4(3x + 6)

Answer 2

what do you notice about equations that are in each of these categories? use complete sentences to answer the questions that follow