Question 138·Hard·Systems of Two Linear Equations in Two Variables

Once you see $3x-7y$ and $x+4y$ repeating, try letting $A=3x-7y$ and $B=x+4y$, and rewrite each equation using only $A$ and $B$.

After rewriting, you should have two linear equations in $A$ and $B$. Use substitution or elimination to find $A$, and remember that $A$ represents $3x-7y$.

In Desmos, enter the first equation exactly as written: 2(3x-7y)+5(x+4y)=18. On the next line, enter the second equation: 4(3x-7y)-2(x+4y)=6. Desmos will draw two straight lines (implicit relations) representing these equations.

Zoom or pan until you can see where the two graphs intersect. Tap or click on the intersection point; Desmos will display its coordinates $(x,y)$.

In a new expression line, type 3*(x-coordinate) - 7*(y-coordinate) using the numbers from the intersection point. The value that Desmos outputs is the value of $3x-7y$ for the solution to the system.

Notice that $3x-7y$ and $x+4y$ appear in both equations. To simplify the system, let $A=3x-7y$ and $B=x+4y$ so we can work with $A$ and $B$ instead of $x$ and $y$.

Substitute $A$ and $B$ into each equation. From the first equation: $2(3x-7y)+5(x+4y)=18$ becomes $2A+5B=18$. From the second equation: $4(3x-7y)-2(x+4y)=6$ becomes $4A-2B=6$. To make numbers smaller, divide this second equation by 2 to get an equivalent equation $2A-B=3$.

From $2A-B=3$, solve for $B$: add $-2A$ to both sides to get $-B=3-2A$, so $B=2A-3$. Substitute this into the first equation $2A+5B=18$: you get $2A+5(2A-3)=18$. Distribute the 5: $2A+10A-15=18$. Combine like terms: $12A-15=18$. Add 15 to both sides to obtain $12A=33$.

From $12A=33$, divide both sides by 12 to get $A=\dfrac{33}{12}$. Simplify by dividing numerator and denominator by 3: $A=\dfrac{11}{4}$. Since $A=3x-7y$, the value of $3x-7y$ is $\dfrac{11}{4}$, which corresponds to choice B.