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.

In Desmos, type the two equations exactly as given:

• (3x+4y)/7 - (2x-5y)/3 = 1
• (3x+4y)/7 + (2x-5y)/3 = 29

Desmos will graph these as two lines in the $xy$-plane.

Click on the point where the two lines intersect. Desmos will display the coordinates $(x,y)$ of this intersection; these are the solution values for $x$ and $y$.

In a new Desmos line, type 5*(x-value) - (y-value) using the $x$ and $y$ from the intersection point you found (for example, if the intersection is (a,b), type 5*a - b). The resulting output is the value of $5x - y$.

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.