Questions and Answers
What is the best interpretation of the solution when a linear equation results in a variable expression set equal to the same variable expression?
The equation has infinite solutions.
How many solutions exist for the given equation: $\frac{1}{2}(x + 12) = 4x - 1$?
one
What is the value of x in the equation $\frac{1}{2}x (x - 14) + 11 = \frac{1}{2}x - (x - 4)$?
0
What is the value of x in the equation $1.5(x + 4) - 3 = 4.5(x - 2)$?
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How many solutions exist for the given equation: $12x + 1 = 3(4x + 1) - 2$?
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Solve for x in the equation $9(x + 1) = 25 + x$.
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Which equation has no solution?
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Solve for n in the equation $n + 1 = 4(n - 8)$.
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How many solutions exist for the equation: $0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20)$?
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How many solutions exist for the equation: $3(x - 2) = 22 - x$?
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Solve for x in the equation $6(x - 1) = 9(x + 2)$.
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What is the best interpretation when the equation $\frac{2}{3}(6x - 3) = \frac{1}{2}(6x - 4)$ simplifies to $4x = 3x$?
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What is the value of x in the equation $2.5(6x - 4) = 10 + 4(1.5 + 0.5x)$?
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How can Lily's partial solution from the equation $4(x - 1) - x = 3(x + 5) - 11$ be interpreted?
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What is the value of n in the equation $\frac{1}{2}(n - 4) - 3 = 3 - (2n + 3)$?
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How many solutions exist for the equation $3x + 13 = 3(x + 6) + 1$?
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What is the value of x in the equation $1.5(x + 4) - 3 = 4.5(x - 2)$?
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Solve for x in the equation $5(x - 10) = 30 - 15x$.
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Study Notes
Infinite Solutions
- If a linear equation simplifies to the same variable expression on both sides, it has infinite solutions.
Unique Solutions
- The equation ( \frac{1}{2}(x + 12) = 4x - 1 ) has one solution.
- The equation ( 0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20) ) results in one solution.
- The equation ( 3(x - 2) = 22 - x ) also results in one solution.
Solution Equations
- For the equation ( \frac{1}{2}x(x - 14) + 11 = \frac{1}{2}x - (x - 4) ), the solution leads to ( x = 0 ) after simplification.
- The value of ( x ) in the equation ( 1.5(x + 4) - 3 = 4.5(x - 2) ) is found to be 4.
- Solving ( 9(x + 1) = 25 + x ) results in ( x = 2 ).
- ( n + 1 = 4(n - 8) ) simplifies to ( n = 11 ).
No Solution
- The equation ( 5 + 2(3 + 2x) = x + 3(x + 1) ) has no solution.
- Interpretation of ( 4(x - 1) - x = 3(x + 5) - 11 ) indicates no solution.
Solutions Leading to Contradictions
- The equation ( 6(x - 1) = 9(x + 2) ) simplifies to ( x = -8 ).
- The equation ( 3x + 13 = 3(x + 6) + 1 ) yields zero solutions and indicates a contradiction.
Additional Key Values
- For the equation ( 1.5(x + 4) - 3 = 4.5(x - 2) ), the solution for ( x ) is confirmed as 4 again.
- In ( 5(x - 10) = 30 - 15x ), solving gives ( x = 4 ).
Summary of Solutions
- Equations can lead to infinite solutions, unique solutions, or no solutions based on their structure and simplification. Ensure to analyze the equations for potential contradictions or consistent identities.
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