First exercise
Given
[tex]\dfrac{3x+1}{x+4}=\dfrac{2}{3}[/tex]
We can cross multiply this equation to get
[tex]3(3x+1)=2(x+4)[/tex]
Expand both terms:
[tex]9x+3=2x+8[/tex]
Subtract 2x from both sides:
[tex]7x+3=8[/tex]
Subtract 3 from both sides:
[tex]7x=5[/tex]
Divide both sides by 7:
[tex]x=\dfrac{5}{7}[/tex]
Second exercise
Let [tex]p,o,a[/tex] be the number of pine, oath and ash trees. We are given:
[tex]\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}[/tex]
We can substitute the expression for p of the first equation in the last equation to get
[tex]p+o+a=300\iff \dfrac{5o}{8}+o+a=300 \iff \dfrac{13}{8}o+a=300[/tex]
Now we use the second equation, to express o in terms of a:
[tex]\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[/tex]
