Question

The graph of a line in the $xy$-plane has slope $2$ and contains the point $(1,8)$. The graph of a second line passes through the points $(1,2)$ and $(2,1)$. If the two lines intersect at the point $(a,b)$, what is the value of $a+b$ ?

• A. 4
• B. 3
• C. -1
• D. -4

Answer 1

Answer: (B)

3

First line has slope $2$ and passes through the point $(1,8)$

If the equation of the line is of the form $y=mx+c$, we have $m=2$

Also, since it passes through $(1,8),8=2\left(1\right)+c,\Rightarrow c=6$

The first line's equation is thus $y=2x+6$

For the second line, we use $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$

$\Rightarrow \frac{y-2}{x-1}=\frac{1-2}{2-1}$

$\Rightarrow \frac{y-2}{x-1}=-1$

$\Rightarrow y-2=1-x$ or $x+y=3$

Substituting $y=3-x$ in first line's equation, we get

$3-x=2x+6$

$\therefore 3x=-3$ or $x=-1$

$\Rightarrow y=3-(-1)=4$

Thus, the intersection point becomes $(-1,4)$ and so $-1+4=3$