SAT Question 58477a6c: Systems of Linear Equations answer and explanation with Desmos solution

Choice C is correct. A system of two linear equations in two variables has at least one solution if the equations represent the same line or lines that intersect at exactly one point. Statement I gives the equation $3x+13.5y=10.5$. Multiplying both sides of the given equation, $2x+9y=7$, by $\frac{3}{2}$ yields $3x+13.5y=10.5$, so the equation in statement I is equivalent to the given equation. Therefore, the system formed by these two equations represents the same line and has infinitely many solutions. Statement II gives the equation $3x-13.5y=10.5$. Adding the right- and left-hand sides of $3x+13.5y=10.5$ and $3x-13.5y=10.5$ yields $3x+13.5y+3x-13.5y=10.5+10.5$, or $6x=21$. Dividing both sides of this equation by $6$ yields $x=\frac{21}{6}$. Therefore, the lines represented by these equations intersect at exactly one point, where $x=\frac{21}{6}$, and the system formed by the given equation and the equation in statement II has one solution. It follows that both the equation in statement I and the equation in statement II could be the other equation in a system of equations with at least one solution.