Question

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+4, 3x+18=5x+8, 3(4x+8)=4(3x+6)

Answer 1

Let's solve each pair of equations to determine the number of solutions:

1) 2x + 3 = 2x + 7
Subtracting 2x from both sides:
3 = 7
This is a contradiction, so there are no solutions.

2) 7x + 3 = 7x + 3
The equations are identical. They have an infinite number of solutions.

3) 4x + 22 = 8x + 10
Subtracting 4x from both sides:
22 = 4x + 10
Subtracting 10 from both sides:
12 = 4x
Dividing both sides by 4:
3 = x
There is one solution.

4) 5x + 5 = 5x + 4
Subtracting 5x from both sides:
5 = 4
This is a contradiction, so there are no solutions.

5) 3x + 18 = 5x + 4
Subtracting 3x from both sides:
18 = 2x + 4
Subtracting 4 from both sides:
14 = 2x
Dividing both sides by 2:
7 = x
There is one solution.

6) 3x + 18 = 5x + 8
Subtracting 3x from both sides:
18 = 2x + 8
Subtracting 8 from both sides:
10 = 2x
Dividing both sides by 2:
5 = x
There is one solution.

7) 3(4x + 8) = 4(3x + 6)
Expanding the equation:
12x + 24 = 12x + 24
The equations are identical. They have an infinite number of solutions.

So, the pairs of equations that have one solution are 3) 4x + 22 = 8x + 10, 5) 3x + 18 = 5x + 4, and 6) 3x + 18 = 5x + 8.
The pairs of equations that have no solution are 1) 2x + 3 = 2x + 7 and 4) 5x + 5 = 5x + 4.
The pairs of equations that have an infinite number of solutions are 2) 7x + 3 = 7x + 3 and 7) 3(4x + 8) = 4(3x + 6).