To make [tex]\( r \)[/tex] the subject of the equation [tex]\( m = \sqrt{\frac{6a + r}{5r}} \)[/tex], follow these steps:
1. Square both sides of the equation to eliminate the square root:
[tex]\[
m^2 = \left(\sqrt{\frac{6a + r}{5r}}\right)^2
\][/tex]
This simplifies to:
[tex]\[
m^2 = \frac{6a + r}{5r}
\][/tex]
2. Multiply both sides by [tex]\( 5r \)[/tex] to clear the fraction:
[tex]\[
5r \cdot m^2 = 6a + r
\][/tex]
This can be written as:
[tex]\[
5m^2r = 6a + r
\][/tex]
3. Rearrange the equation to collect all terms involving [tex]\( r \)[/tex] on one side:
[tex]\[
5m^2r - r = 6a
\][/tex]
4. Factor out the common [tex]\( r \)[/tex] on the left-hand side:
[tex]\[
r(5m^2 - 1) = 6a
\][/tex]
5. Solve for [tex]\( r \)[/tex] by dividing both sides by [tex]\( 5m^2 - 1 \)[/tex]:
[tex]\[
r = \frac{6a}{5m^2 - 1}
\][/tex]
Thus, the expression for [tex]\( r \)[/tex] in terms of [tex]\( m \)[/tex] and [tex]\( a \)[/tex] is:
[tex]\[
r = \frac{6a}{5m^2 - 1}
\][/tex]
