Let's analyze the steps taken to solve the equation \( 12.5x − 10.2 = 3(2.5x + 4.2) - 6 \) and identify the properties or procedures used in each step.
Step 1:
\( 12.5x − 10.2 = 7.5x + 12.6 − 6 \)
Analysis:
Here, the right-hand side of the equation is simplified. The expression \(3(2.5x + 4.2)\) is distributed:
- Multiply \(3\) by \(2.5x\) to get \(7.5x\).
- Multiply \(3\) by \(4.2\) to get \(12.6\).
- From \(12.6\), subtract \(6\) to get \(6.6\).
Property used: Distributive Property and Simplification (combining like terms).
Step 2:
\( 12.5x − 10.2 = 7.5x + 6.6 \)
Analysis:
This step involves simplifying further:
- Combine \(12.6 - 6\) on the right-hand side to give \(6.6\).
Property used: Combining Like Terms.
Step 3:
\( 12.5x = 7.5x + 16.8 \)
Analysis:
In this step, we add \(10.2\) to both sides of the equation to isolate terms involving \(x\):
- \(12.5x - 10.2 + 10.2 = 7.5x + 6.6 + 10.2\).
Property used: Addition Property of Equality (adding the same value to both sides).
Step 4:
\( 5x = 16.8 \)
Analysis:
To isolate \(x\), we subtract \(7.5x\) from both sides:
- \(12.5x - 7.5x = 16.8\).
Property used: Subtraction Property of Equality (subtracting the same value from both sides).
Step 5:
\( x = 3.36 \)
Analysis:
Finally, we divide both sides by \(5\) to solve for \(x\):
- \(5x / 5 = 16.8 / 5\).
Property used: Division Property of Equality (dividing both sides by the same non-zero number).
In summary, each step utilized fundamental algebraic properties: the distributive property, combining like terms, and equality properties (addition, subtraction, division) to manipulate the equation until the variable \(x\) was isolated.
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