To solve the equation [tex]\(\frac{2}{3} x - \frac{1}{9} x + 5 = 20\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Combine like terms involving [tex]\(x\)[/tex]:
The given equation is:
[tex]\[
\frac{2}{3} x - \frac{1}{9} x + 5 = 20
\][/tex]
To combine [tex]\(\frac{2}{3} x\)[/tex] and [tex]\(\frac{1}{9} x\)[/tex], first find a common denominator. The common denominator for 3 and 9 is 9. Rewrite each fraction with this common denominator:
[tex]\[
\frac{2}{3} x = \frac{6}{9} x
\][/tex]
[tex]\[
\frac{6}{9} x - \frac{1}{9} x = \frac{5}{9} x
\][/tex]
So the equation now is:
[tex]\[
\frac{5}{9} x + 5 = 20
\][/tex]
2. Isolate the term with [tex]\(x\)[/tex]:
Subtract 5 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
\frac{5}{9} x + 5 - 5 = 20 - 5
\][/tex]
Simplifying this gives:
[tex]\[
\frac{5}{9} x = 15
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], multiply both sides of the equation by the reciprocal of [tex]\(\frac{5}{9}\)[/tex], which is [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
x = 15 \times \frac{9}{5}
\][/tex]
Simplifying the right-hand side:
[tex]\[
x = 15 \times \frac{9}{5} = 15 \times 1.8 = 27
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = 27
\][/tex]
