Question 689·Hard·Algebra

Type both equations as written into Desmos, one per line, and click where their lines cross. Desmos shows $(1.5,0.25)$. An intersection lies on both lines, so the pair makes both equations true: $x=1.5$ and $y=0.25$.

Use the coordinates in $3x-7y$, not in the other repeated group. Type $3(1.5)-7(0.25)$ on a new Desmos line. It shows $2.75$; tap the fraction button to see $\frac{11}{4}$. The point solves the system, but the requested expression is what you answer. So $3x-7y=\frac{11}{4}$. Choice B.

The same two groups appear in both equations. Use substitution to name them $A=3x-7y$ and $B=x+4y$. Replacing each whole group turns the system into a shorter one, and the question asks for $A$:

To use elimination, make the $B$-terms opposites. Multiply every term of $2A+5B=18$ by $2$:

Multiply every term of $4A-2B=6$ by $5$, including its right side: $20A-10B=30$. Now its $-10B$ can cancel the $+10B$ from the previous step.

Add the two new equations, so the opposite $B$-terms cancel: $(4A+10B)+(20A-10B)=36+30$. Combine like terms:

Divide both sides by $24$: $A=\frac{66}{24}$. You have $A$ directly, so there’s no need to find $x$, $y$, or $B$.

Divide the numerator and denominator by their common factor $6$: $A=\frac{11}{4}$. Since $A$ means $3x-7y$, the requested value is $3x-7y=\frac{11}{4}$. Choice B.