Substitute -3 for a and 8 for b in the given equations.
Tell whether each statement is True or False after substituting for a and b.
A. The equation 5x-7(x-1)= ax+b has exactly one solution. T or F
B. The equation 3(x-5)-7=ax+b has no solution. T or F
C. The equation 2-7x+3+4x=ax+b has no solution. T or F
D. The equation -3(x-3)-1=ax+b has infinitely many solutions. T or F
E. The equation -5x+2+2x+4=ax+b has infinitely many solutions. T or F
PLEASE HELP ME!!! ( and thank you so much if you do :P )
thank you so much!! this was super helpful!!
To determine whether each statement is true or false after substituting for a and b in the given equations, we substitute the values -3 for a and 8 for b.
A. The equation 5x - 7(x - 1) = ax + b
Substituting -3 for a and 8 for b:
5x - 7(x - 1) = -3x + 8
5x - 7x + 7 = -3x + 8
-2x + 7 = -3x + 8
Now, we solve for x:
-2x + 7 = -3x + 8
x -3x = 8 - 7
-2x = 1
x = -1/2
The equation has exactly one solution. Therefore, the statement is True.
B. The equation 3(x - 5) - 7 = ax + b
Substituting -3 for a and 8 for b:
3(x - 5) - 7 = -3x + 8
3x - 15 - 7 = -3x + 8
3x - 22 = -3x + 8
Now, we solve for x:
3x - 22 = -3x + 8
6x - 22 = 8
6x = 30
x = 5
The equation has a unique solution. Therefore, the statement is False.
C. The equation 2 - 7x + 3 + 4x = ax + b
Substituting -3 for a and 8 for b:
2 - 7x + 3 + 4x = -3x + 8
-3x + 5 = -3x + 8
The equation simplifies to:
5 = 8
This is not possible unless there is a mistake in the calculation. Therefore, the statement is False.
D. The equation -3(x - 3) - 1 = ax + b
Substituting -3 for a and 8 for b:
-3(x - 3) - 1 = -3x + 8
-3x + 9 - 1 = -3x + 8
-3x + 8 = -3x + 8
The equation simplifies to:
0 = 0
This is true because both sides of the equation are equal. Therefore, the statement is True.
E. The equation -5x + 2 + 2x + 4 = ax + b
Substituting -3 for a and 8 for b:
-5x + 2 + 2x + 4 = -3x + 8
-3x + 6 = -3x + 8
The equation simplifies to:
6 = 8
This is not possible unless there is a mistake in the calculation. Therefore, the statement is False.
To summarize:
A. The equation 5x - 7(x - 1) = ax + b has exactly one solution.
The statement is True.
B. The equation 3(x - 5) - 7 = ax + b has no solution.
The statement is False.
C. The equation 2 - 7x + 3 + 4x = ax + b has no solution.
The statement is False.
D. The equation -3(x - 3) - 1 = ax + b has infinitely many solutions.
The statement is True.
E. The equation -5x + 2 + 2x + 4 = ax + b has infinitely many solutions.
The statement is False.
I will do two of them, you do the rest in the same way
A) if a=-3 and b = 8 , you have
5x-7(x-1)= -3x+8
5x - 7x + 7 = -3x + 8
x = 1 , so it looks like one solution
d) -3(x-3)-1=ax+b
-3(x-3)-1= -3x+8
-3x + 9 - 1 = -3x + 8
0 = 0
If your variables drop out and you end up with a true statement, then you
have an infinite number of solutions
If your variables drop out and you end up with a false statement, then you
have no solution.
Of course if you can solve the equation, like in A), you will have one solution.