Question

what is the solution to the equation?
$13\frac{3}{4}+x=7\frac{1}{4}$
$\circ\\ x=-6\frac{1}{4}$
$\circ\\ x=-6\frac{1}{2}$
$\circ\\ x=20$
$\circ\\ x=21$

Explanation:

Step1: Isolate the variable \( x \)

To solve for \( x \) in the equation \( 13\frac{3}{4} + x = 7\frac{1}{4} \), we subtract \( 13\frac{3}{4} \) from both sides of the equation. This gives us \( x = 7\frac{1}{4} - 13\frac{3}{4} \).

Step2: Convert mixed numbers to improper fractions (optional, but helpful)

First, convert \( 7\frac{1}{4} \) to an improper fraction: \( 7\frac{1}{4}=\frac{7\times4 + 1}{4}=\frac{29}{4} \). Then, convert \( 13\frac{3}{4} \) to an improper fraction: \( 13\frac{3}{4}=\frac{13\times4+3}{4}=\frac{55}{4} \). Now the equation becomes \( x=\frac{29}{4}-\frac{55}{4} \).

Step3: Subtract the fractions

When subtracting fractions with the same denominator, we subtract the numerators: \( x=\frac{29 - 55}{4}=\frac{-26}{4} \).

Step4: Simplify the fraction

Simplify \( \frac{-26}{4} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So \( \frac{-26\div2}{4\div2}=\frac{-13}{2} \). Convert \( \frac{-13}{2} \) back to a mixed number: \( -6\frac{1}{2} \).

Answer:

\( x = -6\frac{1}{2} \) (corresponding to the option \( x=-6\frac{1}{2} \))