To make [tex]\( q \)[/tex] the subject of the formula [tex]\( 2(q + p) = 1 + 5q \)[/tex], follow these steps:
1. Expand and simplify the equation:
Start with the equation:
[tex]\[
2(q + p) = 1 + 5q
\][/tex]
Distribute the 2 on the left-hand side:
[tex]\[
2q + 2p = 1 + 5q
\][/tex]
2. Isolate the terms involving [tex]\( q \)[/tex]:
Move all terms involving [tex]\( q \)[/tex] to one side of the equation. Subtract [tex]\( 2q \)[/tex] from both sides:
[tex]\[
2p = 1 + 5q - 2q
\][/tex]
Simplify the right-hand side:
[tex]\[
2p = 1 + 3q
\][/tex]
3. Solve for [tex]\( q \)[/tex]:
Isolate [tex]\( q \)[/tex] by subtracting 1 from both sides:
[tex]\[
2p - 1 = 3q
\][/tex]
Finally, divide both sides by 3 to solve for [tex]\( q \)[/tex]:
[tex]\[
q = \frac{2p - 1}{3}
\][/tex]
Thus, the solution for [tex]\( q \)[/tex] in terms of [tex]\( p \)[/tex] is:
[tex]\[
q = \frac{2p - 1}{3}
\][/tex]
And as requested, in the form [tex]\(\frac{a p - b}{c}\)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are positive integers:
[tex]\[
q = \frac{2p - 1}{3}
\][/tex]
Here, [tex]\( a = 2 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = 3 \)[/tex].
