determine whether the equation is a conditional equation, an identity, or a contradiction.
3x5x=2x+7
3(x+1)3x-7
3x5x = 2x + 722. 3(x+1)= 3x-7
The Correct Answer and Explanation is :
Let’s analyze the given equations and determine whether they are conditional equations, identities, or contradictions.
Equation 1: (3x5x = 2x + 7)
This equation seems to have a typographical issue. If interpreted as:
[
3x + 5x = 2x + 7
]
Simplifying:
[
8x = 2x + 7
]
Subtract (2x) from both sides:
[
6x = 7
]
Divide by 6:
[
x = \frac{7}{6}
]
Since we obtained a specific value for (x), this equation is a conditional equation because it is only true for (x = \frac{7}{6}).
Equation 2: (3(x+1) = 3x – 7)
Expanding:
[
3x + 3 = 3x – 7
]
Subtract (3x) from both sides:
[
3 = -7
]
This is a false statement, meaning the equation has no solution. Since no value of (x) satisfies the equation, it is a contradiction.
Summary of Classification
- (3x + 5x = 2x + 7) → Conditional equation (true for (x = \frac{7}{6})).
- (3(x+1) = 3x – 7) → Contradiction (no solution).
Explanation
- Conditional Equations: These equations are true for some values of the variable but not all. They have one or more specific solutions.
- Identities: These equations are always true for all values of the variable (e.g., (x + x = 2x)).
- Contradictions: These equations are never true; they result in a false statement (e.g., (2 = 3)).
Now, let’s generate an image illustrating the solution visually.
The image visually represents the solutions to the given algebraic equations. It includes step-by-step simplifications for both equations, classifying one as a conditional equation and the other as a contradiction. Let me know if you need further modifications or explanations!
