Isolating a variable in an equation involves manipulating the equation to get the variable by itself on one side. Here are the general steps to isolate a variable:
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Identify the variable: Determine which variable you want to isolate.
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Simplify each side: If there are any like terms on either side, combine them. Also, simplify expressions where possible.
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Add or subtract: If the variable is being added or subtracted, do the opposite operation to both sides of the equation to move the variable terms to one side. For example:
- If the variable is \( x \) and the equation is \( x + 5 = 12 \), you subtract 5 from both sides: \[ x + 5 - 5 = 12 - 5 \] \[ x = 7 \]
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Multiply or divide: If the variable is multiplied or divided by a coefficient, perform the opposite operation to both sides to isolate the variable:
- If the equation is \( 3x = 12 \), divide both sides by 3: \[ \frac{3x}{3} = \frac{12}{3} \] \[ x = 4 \]
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Handle fractions: If the variable is part of a fraction, you might multiply both sides by the denominator to eliminate the fraction:
- For example, in \( \frac{x}{3} = 5 \), multiply both sides by 3: \[ 3 \cdot \frac{x}{3} = 3 \cdot 5 \] \[ x = 15 \]
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Use the order of operations: If there are multiple operations involved, follow the order of operations (PEMDAS/BODMAS) to simplify step-by-step.
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Check your work: Substitute your isolated variable back into the original equation to ensure it satisfies the equation.
By following these steps, you can systematically isolate a variable in an equation. If you have a specific equation you're working with, feel free to share it for a step-by-step demonstration!
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