Let the numerator and denominator be x and y respectively
First statement: The denominator of a fraction is two more than three times the numerator
[tex]y=\text{ 3x + 2}[/tex]Second statement: . If both numerator and denominator are decreased by two, the simplified result is 2/7.
[tex]\frac{x-2}{y-2}\text{ = }\frac{2}{7}[/tex]From the equations we have derived, we can now solve for x and y
From the second equation:
[tex]\begin{gathered} \frac{x-2}{y-2}\text{ = }\frac{2}{7} \\ Cross-Multiply \\ 2(y-2)\text{ = 7\lparen x-2\rparen} \\ 2y\text{ - 4 = 7x - 14} \end{gathered}[/tex]Substitute the expression for y from the first equation and solve for x:
[tex]\begin{gathered} 2(3x\text{ + 2\rparen - 4= 7x -14} \\ 6x\text{ + 4 - 4 = 7x - 14} \\ Collect\text{ like terms} \\ 6x\text{ - 7x = -14} \\ -x\text{ = -14} \\ x\text{ = 14} \end{gathered}[/tex]Substitute the value of x into the expression for y:
[tex]\begin{gathered} y\text{ = 3x + 2} \\ y\text{ = 3\lparen14\rparen + 2} \\ y\text{ = 44} \end{gathered}[/tex]Hence, the original fraction was 14/44
