Question

Determine which pairs of equations have one solution no solution infinite solutions 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 analyze each pair of equations:

1) 2x+3=2x+7
This equation is inconsistent and has no solution. When we simplify it, we get:
2x+3 = 2x+7
-2x -2x
3 = 7
Since 3 is not equal to 7, there is no solution.

2) 7x+3=7x+3
This equation is always true and has infinite solutions. When we simplify it, we get:
7x+3 = 7x+3
-7x -7x
3 = 3
Since 3 is equal to 3, the equation is true for all values of x.

3) 4x+22=8x+10
This equation is consistent and has one solution. When we simplify it, we get:
4x+22 = 8x+10
-4x -4x
22 = 4x+10
-10 -10
12 = 4x
3 = x
So the solution is x = 3.

4) 5x+5=5x+4
This equation is inconsistent and has no solution. When we simplify it, we get:
5x+5 = 5x+4
-5x -5x
5 = 4
Since 5 is not equal to 4, there is no solution.

5) 3x+18=5x+8
This equation is consistent and has one solution. When we simplify it, we get:
3x+18 = 5x+8
-3x -3x
18 = 2x+8
-8 -8
10 = 2x
5 = x
So the solution is x = 5.

6) 3(4x+8)=4(3x+6)
This equation is consistent and has one solution. When we simplify it, we get:
12x+24 = 12x+24
The equation simplifies to 24 = 24, which is always true. This means that the equation is true for all values of x.

To summarize:
- Equations 2) and 6) have infinite solutions.
- Equations 1), 4), and 7) have no solution.
- Equations 3) and 5) have one solution.