SAT Question bbb0359a: Linear Equations In Two Variables answer and explanation with Desmos solution

Choice A is correct. The linear relationship between $x$ and $y$ can be represented by the equation $y=mx+b$, where $m$ is the slope of the line in the xy-plane that represents the relationship, and $b$ is the y-coordinate of the y-intercept. The slope can be computed using any two points on the line. The slope of a line between any two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, on the line can be calculated using the slope formula, $m=\frac{y_2-y_1}{x_2-x_1}$. In the given table, each value of $x$ and its corresponding value of $y$ can be represented by a point $\left(x,y\right)$. In the given table, when the value of $x$ is $1$, the corresponding value of $y$ is $11$ and when the value of $x$ is $2$, the corresponding value of $y$ is $16$. Therefore, the points $\left(1,11\right)$ and $\left(2,16\right)$ are on the line. Substituting $\left(1,11\right)$ and $\left(2,16\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{16-11}{2-1}$, or $m=5$. Substituting $5$ for $m$ in the equation $y=mx+b$ yields $y=5x+b$. Substituting the first value of $x$ in the table, $1$, and its corresponding value of $y$, $11$, for $x$ and $y$, respectively, in this equation yields $11=5\left(1\right)+b$, or $11=b+5$. Subtracting $5$ from both sides of this equation yields $6=b$. Substituting $6$ for $b$ in the equation $y=5x+b$ yields $y=5x+6$. Therefore, the equation $y=5x+6$ represents the linear relationship between $x$ and $y$.