What is the slope of the graph of 10x - 5y = -12 in the xy-plane?

1. TRANSLATE the problem information

• Given: Linear equation \(10x - 5y = -12\)
• Find: The slope of this line

2. INFER the solution strategy

• To find slope, I need the equation in slope-intercept form: \(y = mx + b\)
• The coefficient m will be the slope
• Current equation is in standard form, so I need to solve for y

3. SIMPLIFY to isolate the y-term

• Start with: \(10x - 5y = -12\)
• Subtract 10x from both sides: \(-5y = -10x - 12\)

4. SIMPLIFY to solve for y

• Divide everything by -5: \(y = \frac{-10x - 12}{-5}\)
• Distribute the division: \(y = \frac{-10x}{-5} + \frac{-12}{-5}\)
• Simplify each term: \(y = 2x + \frac{12}{5}\)

5. INFER the final answer

• The equation \(y = 2x + \frac{12}{5}\) is in slope-intercept form
• The coefficient of x is 2, so the slope is 2

Answer: D. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when dividing by -5

Students correctly identify that they need to divide by -5, but struggle with the negative signs. They might get confused and think:

• \(\frac{-10x}{-5} = -2x\) instead of \(+2x\)
• Or \(\frac{-12}{-5} = -\frac{12}{5}\) instead of \(+\frac{12}{5}\)

This leads to incorrect slopes like -2, causing them to select Choice A (-2).

Second Most Common Error:

Inadequate INFER reasoning: Stopping before fully converting to slope-intercept form

Some students might rearrange to get something like \(-5y = -10x - 12\) and then try to identify slope from this form, not realizing they need to complete the division step. They might incorrectly think the slope is related to the coefficient of x in standard form (10) or get confused about which coefficient represents slope.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically convert between different forms of linear equations while maintaining accuracy with negative signs throughout multi-step algebraic manipulation.