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Determine if 7(x - 2) + 5 = 3(2x - 1) + 1 has no solution, one solution, or infinitely many solutions
Mathematics
Grade 8
Question Content
Tell whether the equation 7(x - 2) + 5 = 3(2x - 1) + 1 has no solution, one solution, or infinitely many solutions.
Correct Answer
One solution: x = 7
Detailed Solution Steps
1
Step 1: Simplify both sides using the distributive property: Left side: 7x - 14 + 5; Right side: 6x - 3 + 1.
2
Step 2: Combine like constants on both sides: Left side: 7x - 9; Right side: 6x - 2.
3
Step 3: Isolate variable terms by subtracting 6x from both sides: 7x - 6x - 9 = 6x - 6x - 2, which simplifies to x - 9 = -2.
4
Step 4: Isolate the constant term by adding 9 to both sides: x - 9 + 9 = -2 + 9, so x = 7. Since we found a single unique value for x, the equation has one 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 one solution for linear equations
A one-variable linear equation has one unique solution when, after simplifying, you can solve for a single value of x. This happens when the coefficients of x on both sides are different (a ≠ c in ax + b = cx + d).
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