Let's solve the equation [tex]\( \frac{3}{4} x + \frac{5}{4} = 4 x \)[/tex].
1. Combine Similar Terms
First, isolate the [tex]\( x \)[/tex]-terms on one side of the equation. Subtract [tex]\( \frac{3}{4} x \)[/tex] from both sides:
[tex]\[
\frac{3}{4} x + \frac{5}{4} - \frac{3}{4} x = 4 x - \frac{3}{4} x
\][/tex]
Simplifying this gives:
[tex]\[
\frac{5}{4} = 4 x - \frac{3}{4} x
\][/tex]
2. Simplify the Equation
Combine the [tex]\( x \)[/tex]-terms on the right-hand side:
[tex]\[
\frac{5}{4} = \left(4 - \frac{3}{4}\right)x
\][/tex]
To combine these, convert 4 to a fraction with a denominator of 4:
[tex]\[
4 x = \frac{16}{4} x
\][/tex]
Thus, the equation becomes:
[tex]\[
\frac{5}{4} = \left(\frac{16}{4} - \frac{3}{4}\right)x
\][/tex]
Simplifying inside the parentheses:
[tex]\[
\frac{5}{4} = \frac{13}{4} x
\][/tex]
3. Solve for [tex]\( x \)[/tex]
To isolate [tex]\( x \)[/tex], multiply both sides by the reciprocal of [tex]\( \frac{13}{4} \)[/tex]:
[tex]\[
x = \frac{5}{4} \times \frac{4}{13}
\][/tex]
Multiplying these fractions together:
[tex]\[
x = \frac{5 \times 4}{4 \times 13} = \frac{20}{52}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{5}{13}
\][/tex]
Hence, the solution to the equation [tex]\( \frac{3}{4} x + \frac{5}{4} = 4 x \)[/tex] is:
[tex]\[ x = \frac{5}{13} \][/tex]
Therefore, the answer is:
[tex]\[ x = \frac{5}{13} \][/tex]
