Determine which pairs of equations have one solution, no solution, and infinite solutions. Two will have one solution, two will have no solution, and two will have an infinite 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

Let's solve each pair of equations:

1) 2x+3=2x+7
Subtract 2x from both sides:
3 = 7
Since 3 does not equal 7, this pair of equations has no solution.

2) 7x+3=7x+3
Subtract 7x from both sides:
3 = 3
Since 3 is equal to 3, this pair of equations has an infinite number of solutions.

3) 4x+22=8x+10
Subtract 4x from both sides:
22 = 4x+10
Subtract 10 from both sides:
12 = 4x
Divide both sides by 4:
3 = x
Since x has a specific value (x = 3), this pair of equations has one solution.

4) 5x+5=5x+4
Subtract 5x from both sides:
5 = 4
Since 5 does not equal 4, this pair of equations has no solution.

5) 3x+18=5x+8
Subtract 3x from both sides:
18 = 2x+8
Subtract 8 from both sides:
10 = 2x
Divide both sides by 2:
5 = x
Since x has a specific value (x = 5), this pair of equations has one solution.

6) 3(4x+8)=4(3x+6)
Distribute the 3 on the left side:
12x+24 = 4(3x+6)
Distribute the 4 on the right side:
12x+24 = 12x+24
Subtract 12x from both sides:
24 = 24
Since 24 is equal to 24, this pair of equations has an infinite number of solutions.

So, the pairs of equations with one solution are:
- 4x+22=8x+10
- 5x+5=5x+4
The pairs with no solution are:
- 2x+3=2x+7
- 5x+5=5x+4
The pairs with an infinite number of solutions are:
- 7x+3=7x+3
- 3(4x+8)=4(3x+6)