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Determine if 6(2x - 7) - 3 = 12x - 21 has no solution, one solution, or infinitely many solutions
Mathematics
Grade 8
Question Content
Tell whether the equation 6(2x - 7) - 3 = 12x - 21 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: 6(2x) - 6(7) - 3 = 12x - 42 - 3.
2
Step 2: Combine like constants on the left side: 12x - 45 = 12x - 21.
3
Step 3: Isolate variable terms by subtracting 12x from both sides: 12x - 12x - 45 = 12x - 12x - 21, which simplifies to -45 = -21.
4
Step 4: Analyze the result. The statement -45 = -21 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
Combining like terms
Like terms are terms with the same variable raised to the same power (or constant terms with no variable). Combining like terms involves adding or subtracting their coefficients to simplify algebraic expressions, a necessary step in solving linear equations.
3
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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