To solve the equation [tex]\(\frac{5}{7} + \frac{2}{x} = \frac{44}{56}\)[/tex], we will follow these steps:
1. Simplify the right-hand side of the equation:
[tex]\[
\frac{44}{56}
\][/tex]
We can simplify [tex]\(\frac{44}{56}\)[/tex] by finding the greatest common divisor (GCD) of 44 and 56, which is 4. Dividing both the numerator and the denominator by 4, we get:
[tex]\[
\frac{44 \div 4}{56 \div 4} = \frac{11}{14}
\][/tex]
So, the equation now becomes:
[tex]\[
\frac{5}{7} + \frac{2}{x} = \frac{11}{14}
\][/tex]
2. Express [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{11}{14}\)[/tex] with a common denominator:
[tex]\[
\frac{5}{7} = \frac{5 \times 2}{7 \times 2} = \frac{10}{14}
\][/tex]
Thus, the equation now looks like:
[tex]\[
\frac{10}{14} + \frac{2}{x} = \frac{11}{14}
\][/tex]
3. Isolate [tex]\(\frac{2}{x}\)[/tex] on one side:
To do this, we subtract [tex]\(\frac{10}{14}\)[/tex] from both sides of the equation:
[tex]\[
\frac{2}{x} = \frac{11}{14} - \frac{10}{14}
\][/tex]
This simplifies to:
[tex]\[
\frac{2}{x} = \frac{1}{14}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we take the reciprocal of [tex]\(\frac{1}{14}\)[/tex]:
[tex]\[
\frac{2}{x} = \frac{1}{14} \implies \frac{2}{x} = \frac{1}{14}
\][/tex]
This implies:
[tex]\[
2 \cdot 14 = x
\][/tex]
Thus:
[tex]\[
x = 28
\][/tex]
Therefore, the solution to the equation [tex]\(\frac{5}{7} + \frac{2}{x} = \frac{44}{56}\)[/tex] is:
[tex]\[
x = 28
\][/tex]
