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Determine if 1/3(3x - 12) = x - 4 has no solution, one solution, or infinitely many solutions
Mathematics
Grade 8
Question Content
Tell whether the equation 1/3(3x - 12) = x - 4 has no solution, one solution, or infinitely many solutions.
Correct Answer
Infinitely many solutions
Detailed Solution Steps
1
Step 1: Simplify the left side using the distributive property: (1/3)(3x) - (1/3)(12) = x - 4.
2
Step 2: Calculate the simplified left side: x - 4 = x - 4.
3
Step 3: Analyze the result. Both sides of the equation are identical. This means every real number value of x will make the equation true, so the equation has infinitely many solutions.
Knowledge Points Involved
1
Distributive property
The distributive property states that a(b + c) = ab + ac, where a, b, c are real numbers. It is used to expand expressions with parentheses, a key step in simplifying linear equations.
2
Identifying infinitely many solutions for linear equations
A one-variable linear equation has infinitely many solutions when simplifying leads to an identical statement on both sides (e.g., x + b = x + b). This occurs when the coefficients of x and the constant terms on both sides are equal (a = c, b = d in ax + b = cx + d), so all real numbers are valid solutions.
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