Alright, let's tackle the problem step-by-step.
### Part (a): Solve the formula [tex]\( T = \frac{w + y + r}{3} \)[/tex] for [tex]\( r \)[/tex].
1. Start with the given formula:
[tex]\[
T = \frac{w + y + r}{3}
\][/tex]
2. To isolate [tex]\( r \)[/tex], first we need to get rid of the denominator, 3. Multiply both sides of the equation by 3:
[tex]\[
3T = w + y + r
\][/tex]
3. Next, isolate [tex]\( r \)[/tex] by subtracting [tex]\( w \)[/tex] and [tex]\( y \)[/tex] from both sides:
[tex]\[
r = 3T - w - y
\][/tex]
So, the formula solved for [tex]\( r \)[/tex] is:
[tex]\[
r = 3T - w - y
\][/tex]
### Part (b): Use the formula [tex]\( r = 3T - w - y \)[/tex] to find the third exam grade.
Given:
- Desired average [tex]\( T = 90 \% \)[/tex]
- First exam grade [tex]\( w = 85 \% \)[/tex]
- Second exam grade [tex]\( y = 89 \% \)[/tex]
Substitute the known values into the formula:
[tex]\[
r = 3 \cdot 90 - 85 - 89
\][/tex]
Let’s break this down step-by-step:
1. Calculate [tex]\( 3 \cdot 90 \)[/tex]:
[tex]\[
3 \cdot 90 = 270
\][/tex]
2. Subtract the first exam grade [tex]\( 85 \% \)[/tex] from [tex]\( 270 \)[/tex]:
[tex]\[
270 - 85 = 185
\][/tex]
3. Subtract the second exam grade [tex]\( 89 \% \)[/tex] from [tex]\( 185 \)[/tex]:
[tex]\[
185 - 89 = 96
\][/tex]
So, the third exam grade [tex]\( r \)[/tex] you need to achieve to have an average of [tex]\( 90 \% \)[/tex] is:
[tex]\[
r = 96 \%
\][/tex]
