Question
Examine the steps used to solve the equation.
12.5x − 10.2 = 3(2.5x + 4.2) - 6
1. 12.5x − 10.2 = 7.5x + 12.6 − 6
2. 12.5x − 10.2 = 7.5x + 6.6
3. 12.5x = 7.5x + 16.8
4. 5x = 16.8
5. x = 3.36
Analyze the steps to determine which properties or procedures were used to complete each step
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
12.5x − 10.2 = 3(2.5x + 4.2) - 6
1. 12.5x − 10.2 = 7.5x + 12.6 − 6
2. 12.5x − 10.2 = 7.5x + 6.6
3. 12.5x = 7.5x + 16.8
4. 5x = 16.8
5. x = 3.36
Analyze the steps to determine which properties or procedures were used to complete each step
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Answers
GPT-4o mini
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.
**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.