To solve the given equation [tex]\( 2p = \frac{2}{3}(q - 5r) \)[/tex] for [tex]\( r \)[/tex] in terms of [tex]\( p \)[/tex] and [tex]\( q \)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
2p = \frac{2}{3}(q - 5r)
\][/tex]
2. Eliminate the fraction by multiplying both sides by 3:
[tex]\[
3 \cdot 2p = 2(q - 5r)
\][/tex]
This simplifies to:
[tex]\[
6p = 2(q - 5r)
\][/tex]
3. Divide both sides by 2 to isolate the terms involving [tex]\( r \)[/tex]:
[tex]\[
3p = q - 5r
\][/tex]
4. Rearrange the equation to solve for [tex]\( r \)[/tex]:
[tex]\[
q - 5r = 3p
\][/tex]
Isolate [tex]\( r \)[/tex] by moving terms involving [tex]\( q \)[/tex] and [tex]\( p \)[/tex]:
[tex]\[
q - 3p = 5r
\][/tex]
Divide both sides by 5:
[tex]\[
r = \frac{q - 3p}{5}
\][/tex]
Thus, [tex]\( r \)[/tex] in terms of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is:
[tex]\[
r = \frac{q - 3p}{5}
\][/tex]
5. Substitute [tex]\( p = 4 \)[/tex] and [tex]\( q = 27 \)[/tex] into the equation to find the value of [tex]\( r \)[/tex]:
[tex]\[
r = \frac{27 - 3 \cdot 4}{5}
\][/tex]
Calculate the expression inside the parentheses:
[tex]\[
27 - 3 \cdot 4 = 27 - 12 = 15
\][/tex]
Now divide by 5:
[tex]\[
r = \frac{15}{5} = 3
\][/tex]
So the value of [tex]\( r \)[/tex] when [tex]\( p = 4 \)[/tex] and [tex]\( q = 27 \)[/tex] is:
[tex]\[
r = 3
\][/tex]
Thus, the final expressions are:
[tex]\[ r = -0.6p + 0.2q \][/tex]
and when [tex]\( p = 4 \)[/tex] and [tex]\( q = 27 \)[/tex], [tex]\( r = 3 \)[/tex].
