To solve the equation [tex]\(\frac{3}{4}\left(\frac{1}{2} x+\frac{2}{3}\right)+\frac{1}{4} x=6 \frac{3}{4}\)[/tex], follow these detailed steps:
1. Simplify Inside the Parentheses:
First, simplify the expression inside the parentheses:
[tex]\[
\frac{1}{2} x + \frac{2}{3}
\][/tex]
2. Distribute [tex]\(\frac{3}{4}\)[/tex] Across the Parentheses:
Apply the distributive property:
[tex]\[
\frac{3}{4} \left( \frac{1}{2} x + \frac{2}{3} \right) = \frac{3}{4} \times \frac{1}{2} x + \frac{3}{4} \times \frac{2}{3}
\][/tex]
Calculate each term:
[tex]\[
\frac{3}{4} \times \frac{1}{2} x = \frac{3}{8} x
\][/tex]
[tex]\[
\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}
\][/tex]
So the equation now looks like:
[tex]\[
\frac{3}{8} x + \frac{1}{2} + \frac{1}{4} x = 6 \frac{3}{4}
\][/tex]
3. Combine Like Terms:
Combine the terms involving [tex]\(x\)[/tex] on the left-hand side:
[tex]\[
\frac{3}{8} x + \frac{1}{4} x = \frac{3}{8} x + \frac{2}{8} x = \frac{5}{8} x
\][/tex]
So the equation becomes:
[tex]\[
\frac{5}{8} x + \frac{1}{2} = 6 \frac{3}{4}
\][/tex]
4. Convert Mixed Number to Improper Fraction:
Convert [tex]\(6 \frac{3}{4}\)[/tex] to an improper fraction:
[tex]\[
6 \frac{3}{4} = \frac{27}{4}
\][/tex]
Now the equation is:
[tex]\[
\frac{5}{8} x + \frac{1}{2} = \frac{27}{4}
\][/tex]
5. Clear the Fraction:
To isolate [tex]\(x\)[/tex], subtract [tex]\(\frac{1}{2}\)[/tex] from both sides. First convert [tex]\(\frac{1}{2}\)[/tex] to a fraction with denominator 4:
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
Subtract this from [tex]\(\frac{27}{4}\)[/tex]:
[tex]\[
\frac{27}{4} - \frac{2}{4} = \frac{25}{4}
\][/tex]
This gives us:
[tex]\[
\frac{5}{8} x = \frac{25}{4}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Multiply both sides by the reciprocal of [tex]\(\frac{5}{8}\)[/tex]:
[tex]\[
x = \frac{25}{4} \times \frac{8}{5}
\][/tex]
Simplify the multiplication:
[tex]\[
x = \frac{25 \times 8}{4 \times 5} = \frac{200}{20} = 10
\][/tex]
Thus, the solution to the equation is:
[tex]\[
x = 10
\][/tex]
