Determine any mistakes and explain how you would fix them.
Understand the Problem
The question is asking to identify any mistakes in the calculations made by Bao and Maia while determining the y-intercept of the line given certain points. Additionally, it requests an explanation on how to fix those mistakes.
Answer
Bao: $b = 8$, Maia's correction: $b = 8$.
Answer for screen readers
Bao's calculation is correct with $b = 8$ while Maia made a mistake and should also have $b = 8$.
Steps to Solve
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Review Bao's Work
- Bao's calculations involve substituting the point $(2, 9)$ into the equation $y = \frac{1}{2}x + b$.
- His substitution gives: $$ 9 = \frac{1}{2}(2) + b $$
- Simplifying the right-hand side: $$ 9 = 1 + b $$
- To find $b$, subtract 1 from both sides: $$ b = 9 - 1 $$ $$ b = 8 $$
- Bao's value for $b$ is correct.
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Review Maia's Work
- Maia substitutes the point $(12, 14)$ into the same equation $y = \frac{1}{2}x + b$.
- Her substitution gives: $$ 14 = \frac{1}{2}(12) + b $$
- Simplifying the right-hand side: $$ 14 = 6 + b $$
- To find $b$, subtract 6 from both sides: $$ b = 14 - 6 $$ $$ b = 8 $$
- Maia incorrectly writes $b = 5$, which is a mistake.
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Identify and Fixing Maia's Mistake
- Maia should have calculated: $$ 14 - 6 = 8 $$
- Clearly, she miscalculated and incorrectly stated $b = 5$.
- The correct conclusion for Maia is: $$ b = 8 $$
Bao's calculation is correct with $b = 8$ while Maia made a mistake and should also have $b = 8$.
More Information
Both students used the correct slope of $\frac{1}{2}$ and substituted the points accurately into the line equation, but Maia made an arithmetic error in her calculations.
Tips
- Confusing operations; Maia incorrectly calculated $14 - 6$.
- Miswriting or misinterpreting variable values during substitution.
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