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Determine if 2/3(6x - 15) = 4x + 2(x - 13) has no solution, one solution, or infinitely many solutions
Mathematics
Grade 8
Question Content
Tell whether the equation 2/3(6x - 15) = 4x + 2(x - 13) has no solution, one solution, or infinitely many solutions.
Correct Answer
One solution: x = 8
Detailed Solution Steps
1
Step 1: Simplify the left side using the distributive property: (2/3)(6x) - (2/3)(15) = 4x - 10.
2
Step 2: Simplify the right side using the distributive property and combine like terms: 4x + 2x - 26 = 6x - 26.
3
Step 3: Set the simplified sides equal: 4x - 10 = 6x - 26.
4
Step 4: Isolate variable terms by subtracting 4x from both sides: 4x - 4x - 10 = 6x - 4x - 26, which simplifies to -10 = 2x - 26.
5
Step 5: Isolate the constant term by adding 26 to both sides: -10 + 26 = 2x - 26 + 26, which simplifies to 16 = 2x.
6
Step 6: Solve for x by dividing both sides by 2: 16/2 = 2x/2, so x = 8. Since we found a single unique value for x, the equation has one solution.
Knowledge Points Involved
1
Distributive property with fractions
The distributive property applies to coefficients that are fractions: (a/b)(cx + d) = (a/b)(cx) + (a/b)(d). It is used to expand expressions with fractional coefficients and parentheses, a key step in simplifying linear equations with fractions.
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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