Question
Question 8 of 18, Step 1 of 1 Determine if the following equation is a conditional equation, an identity, or a contradiction. \[ 8 x-7=3(x-2)+2 \] Answer O Identity O Contradiction
Ask by Schwartz Norton. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation is a conditional equation.
Solution
1. Start with the given equation:
\[
8x - 7 = 3(x - 2) + 2
\]
2. Expand the right-hand side:
\[
3(x - 2) + 2 = 3x - 6 + 2 = 3x - 4
\]
3. The equation becomes:
\[
8x - 7 = 3x - 4
\]
4. Subtract \(3x\) from both sides:
\[
8x - 3x - 7 = -4 \quad \Rightarrow \quad 5x - 7 = -4
\]
5. Add \(7\) to both sides:
\[
5x = 3
\]
6. Divide both sides by \(5\):
\[
x = \frac{3}{5}
\]
Since the equation has exactly one solution, it is a conditional equation.
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Simplify this solution Bonus Knowledge
To determine the nature of the equation \( 8x - 7 = 3(x - 2) + 2 \), let's simplify both sides. If both sides yield the same expression for all values of \( x \), it’s an identity. If it results in a false statement regardless of \( x \), it’s a contradiction. Solving the equation gives us: \[ 8x - 7 = 3x - 6 + 2 \] \[ 8x - 7 = 3x - 4 \] Subtracting \( 3x \) from both sides: \[ 5x - 7 = -4 \] Now, adding 7 to both sides: \[ 5x = 3 \] Thus, \( x = \frac{3}{5} \) is the only solution, indicating it’s a conditional equation. So, the answer would be that this equation is a conditional equation!
