If you let $A = \dfrac{3x+4y}{7}$ and $B = \dfrac{2x-5y}{3}$, rewrite the two equations in terms of $A$ and $B$. What two simple linear equations do you get?

Once you have the system in $A$ and $B$, use adding or subtracting the equations to find $A$ and $B$. Then convert back to find $3x+4y$ and $2x-5y$.

After you know $3x+4y$ and $2x-5y$, think about how to combine these two expressions with addition or subtraction to produce $5x-y$ directly.

Use Desmos's expression list to enter a=(1+29)/2 and b=(29-1)/2. These calculations correspond to adding and subtracting the two equations to determine the repeated fractional expressions.

Enter p=7a and q=3b. The values of $p$ and $q$ represent $3x+4y$ and $2x-5y$, respectively.

Enter p+q. This works because $(3x+4y)+(2x-5y)=5x-y$, so the displayed result verifies the requested value directly.

Notice that the expressions $\dfrac{3x+4y}{7}$ and $\dfrac{2x-5y}{3}$ appear in both equations.

Let

Then the system becomes a much simpler system in $A$ and $B$:

Now solve the system

Add the two equations:

Subtract the first equation from the second (or plug $A=15$ back into one equation):

So

Use the equalities to find $3x+4y$ and $2x-5y$:

From $\dfrac{3x+4y}{7} = 15$,

From $\dfrac{2x-5y}{3} = 14$,

Look at the target expression $5x - y$ and compare it to $3x+4y$ and $2x-5y$.

Add the two equations you just found:

On the left side,

So

Therefore, the value of $5x - y$ is 147.