Question

the equation $\frac{1}{2}(2 + 4x) - 2x = \frac{1}{4}(3x - 7) + \frac{13}{32}$ represents the number of pounds of walnuts, $x$, in a mixture of nuts. solve the equation.
$x = $___

Explanation:

Step1: Expand both sides

$\frac{1}{2}(2) + \frac{1}{2}(4x) - 2x = \frac{1}{4}(3x) - \frac{1}{4}(7) + \frac{13}{32}$
$1 + 2x - 2x = \frac{3x}{4} - \frac{7}{4} + \frac{13}{32}$

Step2: Simplify left side

$1 + (2x - 2x) = \frac{3x}{4} - \frac{7}{4} + \frac{13}{32}$
$1 = \frac{3x}{4} - \frac{7}{4} + \frac{13}{32}$

Step3: Combine constants on right

Convert to 32 denominator: $-\frac{7}{4} = -\frac{56}{32}$
$1 = \frac{3x}{4} - \frac{56}{32} + \frac{13}{32}$
$1 = \frac{3x}{4} - \frac{43}{32}$

Step4: Isolate the x-term

Add $\frac{43}{32}$ to both sides:
$1 + \frac{43}{32} = \frac{3x}{4}$
$\frac{32}{32} + \frac{43}{32} = \frac{3x}{4}$
$\frac{75}{32} = \frac{3x}{4}$

Step5: Solve for x

Multiply both sides by $\frac{4}{3}$:
$x = \frac{75}{32} \times \frac{4}{3}$
$x = \frac{75 \times 4}{32 \times 3} = \frac{300}{96} = \frac{25}{8}$

Answer:

$\frac{25}{8}$ or $3.125$