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Determine if -2(2x + 3) = -4x + 6 has no solution, one solution, or infinitely many solutions
Mathematics
Grade 8
Question Content
Tell whether the equation -2(2x + 3) = -4x + 6 has no solution, one solution, or infinitely many solutions.
Correct Answer
No solution
Detailed Solution Steps
1
Step 1: Simplify the left side using the distributive property: -2(2x) + (-2)(3) = -4x - 6.
2
Step 2: Set the simplified left side equal to the original right side: -4x - 6 = -4x + 6.
3
Step 3: Add 4x to both sides: -4x + 4x - 6 = -4x + 4x + 6, which simplifies to -6 = 6.
4
Step 4: Analyze the result. The statement -6 = 6 is false, so there is no value of x that makes the equation true, meaning the equation has no solution.
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 no solution for linear equations
A one-variable linear equation has no solution when simplifying leads to a false numerical statement (e.g., a = b where a ≠ b). This occurs when the coefficients of x on both sides are equal, but the constant terms are different (a = c, b ≠ d in ax + b = cx + d).
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