First exercise
Given
![\dfrac{3x+1}{x+4}=\dfrac{2}{3}](https://tex.z-dn.net/?f=%5Cdfrac%7B3x%2B1%7D%7Bx%2B4%7D%3D%5Cdfrac%7B2%7D%7B3%7D)
We can cross multiply this equation to get
![3(3x+1)=2(x+4)](https://tex.z-dn.net/?f=3%283x%2B1%29%3D2%28x%2B4%29)
Expand both terms:
![9x+3=2x+8](https://tex.z-dn.net/?f=9x%2B3%3D2x%2B8)
Subtract 2x from both sides:
![7x+3=8](https://tex.z-dn.net/?f=7x%2B3%3D8)
Subtract 3 from both sides:
![7x=5](https://tex.z-dn.net/?f=7x%3D5)
Divide both sides by 7:
![x=\dfrac{5}{7}](https://tex.z-dn.net/?f=x%3D%5Cdfrac%7B5%7D%7B7%7D)
Second exercise
Let
be the number of pine, oath and ash trees. We are given:
![\begin{cases}\frac{p}{o} = \frac{5}{8} \iff p = \frac{5o}{8}\\\frac{o}{a} = \frac{2}{3} \iff o=\frac{2a}{3}\\p+o+a=300\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%5Cfrac%7Bp%7D%7Bo%7D%20%3D%20%5Cfrac%7B5%7D%7B8%7D%20%5Ciff%20p%20%3D%20%5Cfrac%7B5o%7D%7B8%7D%5C%5C%5Cfrac%7Bo%7D%7Ba%7D%20%3D%20%5Cfrac%7B2%7D%7B3%7D%20%5Ciff%20o%3D%5Cfrac%7B2a%7D%7B3%7D%5C%5Cp%2Bo%2Ba%3D300%5Cend%7Bcases%7D)
We can substitute the expression for p of the first equation in the last equation to get
![p+o+a=300\iff \dfrac{5o}{8}+o+a=300 \iff \dfrac{13}{8}o+a=300](https://tex.z-dn.net/?f=p%2Bo%2Ba%3D300%5Ciff%20%5Cdfrac%7B5o%7D%7B8%7D%2Bo%2Ba%3D300%20%5Ciff%20%5Cdfrac%7B13%7D%7B8%7Do%2Ba%3D300)
Now we use the second equation, to express o in terms of a:
![\dfrac{13}{8}o+a=300 \iff \dfrac{13}{8}\cdot \dfrac{2}{3}a+a=300 \iff \dfrac{25}{12}a = 300 \iff a = 300\cdot \dfrac{12}{25} = 144](https://tex.z-dn.net/?f=%5Cdfrac%7B13%7D%7B8%7Do%2Ba%3D300%20%5Ciff%20%5Cdfrac%7B13%7D%7B8%7D%5Ccdot%20%5Cdfrac%7B2%7D%7B3%7Da%2Ba%3D300%20%5Ciff%20%5Cdfrac%7B25%7D%7B12%7Da%20%3D%20300%20%5Ciff%20a%20%3D%20300%5Ccdot%20%5Cdfrac%7B12%7D%7B25%7D%20%3D%20144)