Answer
The equation with no solutions is (C) \( 14x + 23 = 14x - 23 \).
Solution
Solve the equation by following steps:
- step0: Solve for \(x\):
\(14x-23=14x-23\)
- step1: Cancel equal terms:
\(14x=14x\)
- step2: The statement is true:
\(x \in \mathbb{R}\)
Solve the equation \( 14 x+23=14 x-23 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(14x+23=14x-23\)
- step1: Cancel equal terms:
\(23=-23\)
- step2: The statement is false:
\(x \in \varnothing \)
Solve the equation \( -14 x+23=14 x-23 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(-14x+23=14x-23\)
- step1: Move the expression to the left side:
\(-14x-14x=-23-23\)
- step2: Add and subtract:
\(-28x=-23-23\)
- step3: Add and subtract:
\(-28x=-46\)
- step4: Change the signs:
\(28x=46\)
- step5: Divide both sides:
\(\frac{28x}{28}=\frac{46}{28}\)
- step6: Divide the numbers:
\(x=\frac{23}{14}\)
Solve the equation \( -23 x-14=14 x-23 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(-23x-14=14x-23\)
- step1: Move the expression to the left side:
\(-23x-14x=-23+14\)
- step2: Add and subtract:
\(-37x=-23+14\)
- step3: Add and subtract:
\(-37x=-9\)
- step4: Change the signs:
\(37x=9\)
- step5: Divide both sides:
\(\frac{37x}{37}=\frac{9}{37}\)
- step6: Divide the numbers:
\(x=\frac{9}{37}\)
Let's analyze each equation to determine if they have solutions:
(A) \( -23 x - 14 = 14 x - 23 \)
To solve this equation, we can rearrange it:
\[
-23x - 14 - 14x + 23 = 0 \implies -37x + 9 = 0 \implies x = \frac{9}{37}
\]
This equation has a solution.
(B) \( 14 x - 23 = 14 x - 23 \)
This equation simplifies to:
\[
0 = 0
\]
This is true for all values of \( x \), meaning it has infinitely many solutions.
(C) \( 14 x + 23 = 14 x - 23 \)
Rearranging gives:
\[
14x + 23 - 14x + 23 = 0 \implies 46 = 0
\]
This is a contradiction, indicating that this equation has no solutions.
(D) \( -14 x + 23 = 14 x - 23 \)
Rearranging gives:
\[
-14x - 14x + 23 + 23 = 0 \implies -28x + 46 = 0 \implies x = \frac{23}{14}
\]
This equation has a solution.
### Summary of Solutions:
- (A) has a solution.
- (B) has infinitely many solutions.
- (C) has no solutions.
- (D) has a solution.
Thus, the equation that has no solutions is:
**(C)** \( 14 x + 23 = 14 x - 23 \)
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