Respuesta :
Given that a positive interger is 5 less than another integer, we have the equation:
a = b - 5
Where a represents the smaller interger and b represents the larger interger.
Also, the sum of the reciprocal of the smaller interger and twice the reciprocal of the larger is 11/14. we have the equation:
[tex]\frac{1}{a}+(2\times\frac{1}{b})=\frac{11}{14}[/tex]Thus, we have the system of equations:
[tex]\begin{gathered} a=b-5 \\ \\ \frac{1}{a}+\frac{2}{b}=\frac{11}{14} \end{gathered}[/tex]Let's solve for a and b simultaneously using substitution method.
Susbtitue (b - 5) for a in equation 2:
[tex]\frac{1}{b-5}+\frac{2}{b}=\frac{11}{14}[/tex]Solving further:
[tex]\begin{gathered} \frac{1}{b-5}+\frac{2}{b}=\frac{11}{14} \\ \\ \frac{b+2(b-5)}{b(b-5)}=\frac{11}{14} \\ \\ \frac{b+2b-10}{b(b-5)}=\frac{11}{14} \\ \\ \frac{3b-10}{b(b-5)}=\frac{11}{14} \end{gathered}[/tex]Cross multiply:
[tex]\begin{gathered} 14(3b\text{ -10) = 11(b(b-5))} \\ \\ 42b-140=11(b^2-5b) \\ \\ 42b-140=11b^2-55b \\ \\ -140=11b^2-55b-42b \\ \\ 11b^2-55b-42b=-140 \\ \\ 11b^2-97b=-140 \end{gathered}[/tex]Equate to zero:
[tex]11b^2-97b+140=0[/tex]Factor the equation by grouping:
[tex]\begin{gathered} 11b^2-20b-77b+140=0 \\ \\ b(11b-20)-7(11b-20)=0 \\ \\ We\text{ have:} \\ (b-7)(11b-20) \end{gathered}[/tex]Equate each factor to zero and solve for b:
[tex]\begin{gathered} b-7=0 \\ \\ \text{ b=7} \end{gathered}[/tex][tex]\begin{gathered} 11b-20=0 \\ \\ 11b=0+20 \\ \\ 11b=20 \\ \\ b=\frac{20}{11} \end{gathered}[/tex]We have the possible values for:
[tex]b=7,\text{ }\frac{20}{11}[/tex]Substitute 7 and 20/11 for b in equation 1 to find a:
[tex]\begin{gathered} a=b-5 \\ \\ a=7-5 \\ \\ a=2 \end{gathered}[/tex][tex]\begin{gathered} a=b-5 \\ \\ a=\frac{20}{11}-5 \\ \\ a=-\frac{35}{11} \end{gathered}[/tex]Thus, we have:
[tex]\begin{gathered} \text{When a = 2, b = }7 \\ \\ \text{When a = -}\frac{35}{11},b=\frac{20}{11} \end{gathered}[/tex]We are told the value of a is a positive integer, let's take the positive values.
Thus, we have:
a = 2, b = 7
The larger interger is 7, while the smaller interger is 2
ANSWER:
Larger interger = 7
Smaller interger = 2
