Question 1: 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)
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)