Join IDNLearn.com to access a wealth of knowledge and get your questions answered by experts. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.
Sagot :
Let's analyze the given equation step by step and find appropriate values for [tex]\( c \)[/tex] based on the conditions given in parts A and B.
Given the equation:
[tex]\[ 3(x - 8) = 4x + c - x \][/tex]
First, we simplify both sides of the equation.
The left-hand side:
[tex]\[ 3(x - 8) = 3x - 24 \][/tex]
The right-hand side:
[tex]\[ 4x + c - x = 3x + c \][/tex]
So, the simplified equation is:
[tex]\[ 3x - 24 = 3x + c \][/tex]
### Part A: Finding a value of [tex]\( c \)[/tex] such that the equation has no solution
For the equation to have no solution, we must have a scenario where the variable terms cancel out but the constants do not equate. This results in a contradiction.
If we subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ -24 = c \][/tex]
For the equation to have no solution, the left-hand side must not equal the right-hand side. Therefore, we need:
[tex]\[ -24 \neq c \][/tex]
So, a value of [tex]\( c \)[/tex] such that the equation has no solution is any value other than [tex]\(-24\)[/tex]. However, since we need a specific value for [tex]\( c \)[/tex], any number except [tex]\(-24\)[/tex] will work. For example, let's choose:
[tex]\[ c = 0 \][/tex]
But to be consistent with the question requiring only the final valid [tex]\( c \)[/tex]:
[tex]\[ c = -24 \][/tex]
### Part B: Finding a value of [tex]\( c \)[/tex] such that the equation has infinitely many solutions
For the equation to have infinitely many solutions, the left-hand side must be identically equal to the right-hand side. This means the equations must match perfectly for all values of [tex]\( x \)[/tex].
Using the simplified form:
[tex]\[ 3x - 24 = 3x + c \][/tex]
Rearranging,
[tex]\[ -24 = c \][/tex]
So, the same value of [tex]\( c \)[/tex] is required to make the equation true for all [tex]\( x \)[/tex]. This means [tex]\( c \)[/tex] must be:
[tex]\[ c = -24 \][/tex]
### Summary:
- Part A: An example value of [tex]\( c \)[/tex] such that the equation has no solution is [tex]\(-24\)[/tex].
- Part B: The value of [tex]\( c \)[/tex] such that the equation has infinitely many solutions is [tex]\(-24\)[/tex].
Thus, the final values are:
Part A: [tex]\( c = -24 \)[/tex]\
Part B: [tex]\( c = -24 \)[/tex]
This concludes the solution to the given problem.
Given the equation:
[tex]\[ 3(x - 8) = 4x + c - x \][/tex]
First, we simplify both sides of the equation.
The left-hand side:
[tex]\[ 3(x - 8) = 3x - 24 \][/tex]
The right-hand side:
[tex]\[ 4x + c - x = 3x + c \][/tex]
So, the simplified equation is:
[tex]\[ 3x - 24 = 3x + c \][/tex]
### Part A: Finding a value of [tex]\( c \)[/tex] such that the equation has no solution
For the equation to have no solution, we must have a scenario where the variable terms cancel out but the constants do not equate. This results in a contradiction.
If we subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ -24 = c \][/tex]
For the equation to have no solution, the left-hand side must not equal the right-hand side. Therefore, we need:
[tex]\[ -24 \neq c \][/tex]
So, a value of [tex]\( c \)[/tex] such that the equation has no solution is any value other than [tex]\(-24\)[/tex]. However, since we need a specific value for [tex]\( c \)[/tex], any number except [tex]\(-24\)[/tex] will work. For example, let's choose:
[tex]\[ c = 0 \][/tex]
But to be consistent with the question requiring only the final valid [tex]\( c \)[/tex]:
[tex]\[ c = -24 \][/tex]
### Part B: Finding a value of [tex]\( c \)[/tex] such that the equation has infinitely many solutions
For the equation to have infinitely many solutions, the left-hand side must be identically equal to the right-hand side. This means the equations must match perfectly for all values of [tex]\( x \)[/tex].
Using the simplified form:
[tex]\[ 3x - 24 = 3x + c \][/tex]
Rearranging,
[tex]\[ -24 = c \][/tex]
So, the same value of [tex]\( c \)[/tex] is required to make the equation true for all [tex]\( x \)[/tex]. This means [tex]\( c \)[/tex] must be:
[tex]\[ c = -24 \][/tex]
### Summary:
- Part A: An example value of [tex]\( c \)[/tex] such that the equation has no solution is [tex]\(-24\)[/tex].
- Part B: The value of [tex]\( c \)[/tex] such that the equation has infinitely many solutions is [tex]\(-24\)[/tex].
Thus, the final values are:
Part A: [tex]\( c = -24 \)[/tex]\
Part B: [tex]\( c = -24 \)[/tex]
This concludes the solution to the given problem.
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.