Sure, let's solve the equation [tex]\(3x - \frac{1}{4} = \frac{1}{6} - x\)[/tex] step by step using the transposition method.
1. Starting with the given equation:
[tex]\[
3x - \frac{1}{4} = \frac{1}{6} - x
\][/tex]
2. Transpose the term involving [tex]\(x\)[/tex] from the right-hand side to the left-hand side:
[tex]\[
3x + x - \frac{1}{4} = \frac{1}{6}
\][/tex]
Simplify the left-hand side:
[tex]\[
4x - \frac{1}{4} = \frac{1}{6}
\][/tex]
3. Transpose the constant term [tex]\(-\frac{1}{4}\)[/tex] from the left-hand side to the right-hand side:
[tex]\[
4x = \frac{1}{6} + \frac{1}{4}
\][/tex]
4. Find a common denominator to add the fractions on the right-hand side:
[tex]\[
\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}
\][/tex]
So, we have:
[tex]\[
4x = \frac{5}{12}
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides by 4:
[tex]\[
x = \frac{5}{12} \div 4
\][/tex]
Division by 4 is the same as multiplication by [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
x = \frac{5}{12} \times \frac{1}{4} = \frac{5}{48}
\][/tex]
In decimal form:
[tex]\[
x \approx 0.104166666666667
\][/tex]
So, the solution of the equation [tex]\(3 x - \frac{1}{4} = \frac{1}{6} - x\)[/tex] is approximately [tex]\( x = 0.104166666666667 \)[/tex].
