Answer:
No solutions: 6 - 3 + 4x = 4 x + 4
One solution: 6 - 3 + 4x = 3 x + 4
Infinitely many solutions: 6 - 3 + 4x = 4 x + 3
Explanations:
We are to provide a value into the gap, theat will make the equation have:
1. No solutions
2. infinitely many solutions
3. One solution
The given equation is:
6 - 3 + 4x = ...... x + .........
1. No solutions:
Note that for an equation to have no solution, the Right Hand Side should not be equal to the Left Hand Side after simplification.
Therefore, for the equation 6 - 3 + 4x = ...... x + .........
By simplifying the equation:
3 + 4x = ...... x + .........
For the equation to not have solution, put 4 in the first gap, and a number that isn't 4 in the second gap.
3 + 4x = 4x + 4
3 + 4x - 4x = 4
3 = 4 ( False)
Since this is false, the equation 3 + 4x = 4x + 4 is unsolvable:
2. One solution:
For the equation to have one solution, the coefficient of x on the right and left sides must not be the same.
3 + 4x = ...... x + .........
For the equation to have just one solution the coefficient of x on the right hand side must take on a value that is not 4, (e.g. 3), the second gap can take on any value. The constant can take on any value
3 + 4x = 3x + 4
4x - 3x = 4 - 3
x = 1
3. Infinitely many solutions:
Note that for an equation to have infinitely many solutions, the Right Hand Side should be equal to the Left Hand Side after simplification.
6 - 3 + 4x = ...... x + .........
3 + 4x = ...... x + .........
The coefficient of x on the Left hand side has to be 4, and the constant has to be 3 to make the Left Hand Side equal to the Right hand side.
3 + 4x = 4x + 3
3 + 4x - 4x = 3
3 + 0 = 3
3 = 3 (True)
Since LHS = RHS, the given equation has infinitely many solutions.