To solve the equation \( 2x - 5 = 8x + 7 \), we can follow these four steps along with the properties used:
Steps to Solve:
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Isolate the variable on one side: \[ 2x - 5 = 8x + 7 \] Add \( -8x \) and \( 5 \) to both sides: \[ 2x - 8x - 5 + 5 = 7 + 5 \] This simplifies to: \[ -6x = 12 \]
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Multiply or divide to solve for the variable: \[ -6x = 12 \] Divide both sides by \(-6\): \[ x = \frac{12}{-6} \] This simplifies to: \[ x = -2 \]
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Check the solution: Substitute \( x = -2 \) back into the original equation: \[ 2(-2) - 5 = 8(-2) + 7 \] This gives: \[ -4 - 5 = -16 + 7 \] Simplifying both sides: \[ -9 = -9 \] This confirms that our solution is correct.
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State the final answer: Thus, the solution to the equation is: \[ x = -2 \]
Properties Used:
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Addition Property of Equality: If you add (or subtract) the same number to both sides of an equation, the two sides remain equal.
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Multiplication Property of Equality: If you multiply (or divide) both sides of an equation by the same non-zero number, the two sides remain equal.
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Substitution Property: If two values are equal, one can be substituted for the other in any expression or equation.
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Distributive Property (implicitly used when combining like terms): This states that \( a(b + c) = ab + ac \) and helps in combining and simplifying terms in the equation.
These steps and properties lead us to the solution of the equation.
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