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Determine if -3(5x + 9) + 7 = -5(5 + 3x) + 5 has no solution, one solution, or infinitely many solutions
Mathematics
Grade 8
Question Content
Tell whether the equation -3(5x + 9) + 7 = -5(5 + 3x) + 5 has no solution, one solution, or infinitely many solutions.
Correct Answer
Infinitely many solutions
Detailed Solution Steps
1
Step 1: Simplify both sides using the distributive property: Left side: -15x - 27 + 7; Right side: -25 - 15x + 5.
2
Step 2: Combine like constants on both sides: Left side: -15x - 20; Right side: -15x - 20.
3
Step 3: Isolate variable terms by adding 15x to both sides: -15x + 15x - 20 = -15x + 15x - 20, which simplifies to -20 = -20.
4
Step 4: Analyze the result. Wait, correction: Recheck right side simplification: -5(5) + (-5)(3x) +5 = -25 -15x +5 = -15x -20. Left side: -3(5x) -3(9)+7= -15x-27+7=-15x-20. Both sides are identical, so infinitely many solutions. Correction to solution steps: Step 3: After simplifying both sides, we get -15x -20 = -15x -20. Since both sides are identical, every real number x makes the equation true, so there are 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
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 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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