We have to transform each sentence into an algebraic expression.
"The difference between two numbers is 5" can be expressed as
[tex]x-y=5[/tex]"The sum of three times the larger number and twice the smaller number is 30" can be expressed as
[tex]3x+2y=30[/tex]Where x is the larger number, and y is the smaller number.
We form a system of equations with these two
[tex]\left\{ \begin{aligned}x-y=5 \\ 3x+2y=30\end{aligned}\right.[/tex]We can multiply the first equation by 2 and sum them to find the first variable
[tex]\begin{gathered} \left\{ \begin{aligned}2x-2y=10 \\ 3x+2y=30\end{aligned}\right.\rightarrow5x=40 \\ \end{gathered}[/tex]Then, we divide the equation by 5
[tex]\begin{gathered} \frac{5x}{5}=\frac{40}{5} \\ x=8 \end{gathered}[/tex]The larger number is 8.
Now, we find the smaller numbers using the first equation
[tex]\begin{gathered} x-y=5 \\ 8-y=5 \\ -y=5-8 \\ -y=-3 \\ y=3 \end{gathered}[/tex]The smaller number is 3.
