All the points on the line with equation $y=3 x+6$ are translated up 3 units, then translated left 4 units, and then reflected in the line with equation $y=x$. Determine the $y$-intercept of the resulting line.
Solution 1

The original line contains the points $(0,6)$ and $(1,9)$.

Translating these points 3 units up gives the points $(0,9)$ and $(1,12)$.

Translating these points 4 units to the left gives $(-4,9)$ and $(-3,12)$.

Reflecting these points in the line $y=x$ gives $(9,-4)$ and $(12,-3)$.

The slope of the line through these two points is $\frac{-3-(-4)}{12-9}=\frac{1}{3}$.

Substituting $x=9$ and $y=-4$ into $y=\frac{1}{3} x+b$ gives $-4=3+b$ or $b=-7$.

Therefore, the equation of the resulting line is $y=\frac{1}{3} x-7$, so the answer is -7 .

## Solution 2

The equation of the line obtained by translating up by 3 units is $y=3 x+6+3$ or $y=3 x+9$. The equation of the line obtained by translating to the left by 4 units is $y=3(x+4)+9$ or $y=3 x+21$.

To find the equation of the line obtained by reflecting the line in $y=x$, we interchange $x$ and $y$ to get $x=3 y+21$, then solve for $y$ to get $y=\frac{1}{3} x-7$. Therefore, the answer is -7 .

ANSWER: -7