Let's go through each part of the problem step by step.

Part (a) - Solve the formula for $w$:

The average $T$ of three exam grades $y$, $r$, and $w$ is given by the formula: $T = \frac{y + r + w}{3}$ To solve for $w$, follow these steps:

1. Multiply both sides by 3 to eliminate the denominator: $3T = y + r + w$

2. Subtract $y + r$ from both sides to isolate $w$: $w = 3T - y - r$

So, the formula solved for $w$ is: $w = 3T - y - r$

Part (b) - Calculate the required grade on the third exam:

You are given the grades for the first two exams, $y = 85\%$ and $r = 87\%$, and you need to find the grade $w$ on the third exam that would result in an average of $T = 87\%$.

Using the formula derived in part (a): $w = 3T - y - r$

Substitute the known values: $w = 3(87) - 85 - 87$

Now, let's calculate the value of $w$: $w = 261 - 85 - 87 = 261 - 172 = 89$

So, you need to score 89% on the third exam to achieve an average of 87%.

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Would you like any further details or explanations on these steps?

Related Questions:

1. What would the grade $w$ be if the target average $T$ was 90% instead of 87%?
2. How would the result change if your grades were 85%, 88%, and you aimed for an average of 87%?
3. If you got an 89% on the third exam, what would your new average be?
4. How does changing one of the initial grades $y$ or $r$ affect the required grade $w$?
5. What is the minimum grade you could get on the third exam to still achieve at least an 85% average?

Tip:

When solving equations, always perform the same operation on both sides to maintain equality.