SAT Question e25f0807: Linear Functions answer and explanation with Desmos solution

The linear relationship between $x$ and $y$ can be represented by the equation $y=mx+b$, where $m$ and $b$ are constants. It's given in the table that when $x=-12$, $y=-45$. Substituting $-12$ for $x$ and $-45$ for $y$ in the equation $y=mx+b$ yields $-45=-12m+b$, which can be rewritten as $-45+12m=b$. It's also given in the table that when $x=6$, $y=45$. Substituting $6$ for $x$ and $45$ for $y$ in the equation $y=mx+b$ yields $45=6m+b$, which can be rewritten as $45-6m=b$. Substituting $-45+12m$ for $b$ in this equation yields $45-6m=-45+12m$. Adding $6m$ to both sides of this equation yields $45=-45+18m$. Adding $45$ to both sides of this equation yields $90=18m$. Dividing both sides of this equation by $18$ yields $5=m$, or $m=5$. Substituting $5$ for $m$, $-12$ for $x$, and $-45$ for $y$ in the equation $y=mx+b$ yields $-45=5(-12)+b$, or $-45=-60+b$. Adding $60$ to both sides of this equation yields $15=b$. Therefore, $m=5$ and $b=15$. Substituting $5$ for $m$ and $15$ for $b$ in the equation $y=mx+b$ yields $y=5x+15$. Thus, the equation $y=5x+15$ represents the linear relationship between $x$ and $y$. It’s also given that the graph of the linear equation representing this relationship passes through the point $\left(\frac{1}{4},a\right)$. Substituting $\frac{1}{4}$ for $x$ and $a$ for $y$ in the equation $y=5x+15$ yields $a=5\left(\frac{1}{4}\right)+15$, which is equivalent to $a=\frac{5}{4}+15$, or $a=\frac{65}{4}$. Note that 65/4 and 16.25 are examples of ways to enter a correct answer.