Question

find the solution set. then indicate whether the equation is conditional, an identity, or a contradiction.\\(42(x - 1)=-7(8 - x)+35x\\)\
select the correct choice below and fill in any answer boxes present in your choice.\
\\(\boldsymbol{\circ}\\) a. the solution set is \\(\boldsymbol{\square}\\).\
\\(\circ\\) b. the solution is the set of all real numbers.\
\\(\circ\\) c. the solution is the empty set.\
indicate whether the equation is conditional, an identity, or a contradiction.\
choose the correct answer.\
\\(\circ\\) identity\
\\(\circ\\) contradiction\
\\(\circ\\) conditional equation

Explanation:

Step1: Expand both sides

$42x - 42 = -42 + 7x + 35x$

Step2: Simplify right-hand side

$42x - 42 = 42x - 42$

Step3: Isolate variable terms

$42x - 42x = -42 + 42$

Step4: Solve for $x$

$0x = 0$
Wait, correction: Re-express Step1 correctly:

Step1: Expand both sides

$42(x-1) = 42x - 42$; $-7(6-x)+35x = -42 +7x +35x$

Step2: Combine like terms on RHS

$-42 + 42x$

Step3: Set equal and simplify

$42x - 42 = 42x - 42$
This simplifies to $0=0$, which is always true. Wait, correction:
Wait, re-calculate expansion:

Step1: Expand left side

$42(x-1) = 42x - 42$

Step2: Expand right side

$-7(6-x)+35x = -42 +7x +35x = 42x -42$

Step3: Equate and simplify

$42x -42 = 42x -42$
Subtract $42x$ from both sides:
$-42 = -42$
This is a true statement for all real $x$.

Corrected Answer:

B. The solution is the set of all real numbers.
identity

Step1: Expand left-hand side

$42(x-1) = 42x - 42$

Step2: Expand right-hand side

$-7(6-x)+35x = -42 +7x +35x = 42x -42$

Step3: Equate and simplify

$42x - 42 = 42x - 42$

Step4: Verify equality

Subtract $42x$ from both sides: $-42 = -42$
This is always true, so all real numbers are solutions, making it an identity.

Answer:

A. The solution set is $\{1\}$
conditional equation