To solve the equation
[tex]\[
\frac{R}{7} + 7 = \frac{5R}{6},
\][/tex]
we will follow these steps:
1. Isolate the variable on one side: We want to collect all terms involving [tex]\( R \)[/tex] on one side of the equation.
First, subtract [tex]\( \frac{R}{7} \)[/tex] from both sides:
[tex]\[
7 = \frac{5R}{6} - \frac{R}{7}.
\][/tex]
2. Combine the [tex]\( R \)[/tex] terms on the right-hand side of the equation: To do this, we need to have a common denominator. The common denominator of 6 and 7 is 42.
Rewrite each fraction with the common denominator:
[tex]\[
\frac{5R}{6} = \frac{5R \cdot 7}{6 \cdot 7} = \frac{35R}{42},
\][/tex]
[tex]\[
\frac{R}{7} = \frac{R \cdot 6}{7 \cdot 6} = \frac{6R}{42}.
\][/tex]
Now, substitute these in:
[tex]\[
7 = \frac{35R}{42} - \frac{6R}{42}.
\][/tex]
3. Simplify the right-hand side: Combine the fractions.
[tex]\[
7 = \frac{35R - 6R}{42} = \frac{29R}{42}.
\][/tex]
4. Solve for [tex]\( R \)[/tex]: To solve for [tex]\( R \)[/tex], we multiply both sides by 42 to eliminate the fraction:
[tex]\[
7 \times 42 = 29R,
\][/tex]
which simplifies to:
[tex]\[
294 = 29R.
\][/tex]
Next, divide both sides by 29:
[tex]\[
R = \frac{294}{29}.
\][/tex]
So, the solution to the equation is
[tex]\[
R = \frac{294}{29}.
\][/tex]
