SAT Question 18ac8354: Systems of Linear Equations answer and explanation with Desmos solution

It’s given that the graph of line $g$ is shown in the xy-plane and that line $k$ is defined by $165x+py=w$, where $p$ and $w$ are constants. It’s also given that if line $k$ is graphed in this xy-plane, the resulting system of two linear equations will have infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. Therefore, line $g$ and line $k$ have the same slope and the same y-intercept. The slope of a line that passes through the points $(x_1,y_1)$ and $(x_2,y_2)$ can be calculated using the formula $m=\frac{y_2-y_1}{x_2-x_1}$. The graph shows that line $g$ passes through the points $(-10,12)$ and $(0,1)$. Substituting $(-10,12)$ and $(0,1)$ for $(x_1,y_1)$ and $(x_2,y_2)$, respectively, in the formula $m=\frac{y_2-y_1}{x_2-x_1}$ yields $m=\frac{1-12}{0-(-10)}$, or $m=\frac{-11}{10}$. An equation representing a linear relationship can be written in the form $y=mx+b$, where $m$ is the slope and $(0,b)$ is the y-intercept of the line in the xy-plane. Since line $g$ passes through the point $(0,1)$, it follows that $b=1$. Thus, the equation of line $g$ is $y=-\frac{11}{10}x+1$. For line $k$, the equation $165x+py=w$ can be rewritten in $y=mx+b$ form by isolating $y$. Subtracting $165x$ from both sides of this equation yields $py=-165x+w$. Dividing both sides of this equation by $p$ yields $y=\frac{-165}{p}x+\frac{w}{p}$. It follows that in the xy-plane, line $k$ has a slope of $\frac{-165}{p}$ and $\left(0,\frac{w}{p}\right)$ is the y-intercept. Since line $g$ and line $k$ have the same slope and the same y-intercept, it follows that $-\frac{165}{p}=-\frac{11}{10}$ and $\frac{w}{p}=1$. Multiplying both sides of $-\frac{165}{p}=-\frac{11}{10}$ by $-10p$ yields $1,650=11p$. Dividing both sides of this equation by $11$ yields $150=p$. Substituting $150$ for $p$ in the equation $\frac{w}{p}=1$ yields $\frac{w}{150}=1$. Multiplying both sides of this equation by $150$ yields $w=150$. Substituting $150$ for $p$ and $150$ for $w$ in the expression $p+w$ yields $150+150$, which is equivalent to $300$. Therefore, the value of $p+w$ is $300$.