### Instruction: Find the point on the line
\[y = -3x + 5\]that is closest to the point $(-4,-2).$.### Response: I know that the shortest distance from a point to a line is along the perpendicular line that passes through the point.
So I need to find the equation of the line that is perpendicular to $y = -3x + 5$ and passes through $(-4,-2)$.
To find the slope of the perpendicular line, I take the negative reciprocal of the slope of the given line.
The slope of the given line is -3, so the slope of the perpendicular line is 1/3.
Now I can use the point-slope form of a line to write the equation of the perpendicular line.
The point-slope form is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope.
In this case, $(x_1, y_1) = (-4,-2)$ and $m = 1/3$, so the equation is $y - (-2) = (1/3)(x - (-4))$.
Simplifying, I get $y + 2 = (1/3)x + 4/3$.
Subtracting 2 from both sides, I get $y = (1/3)x - 2/3$.
This is the equation of the perpendicular line that passes through $(-4,-2)$.
To find the point of intersection between this line and the given line, I need to solve the system of equations \[\begin{cases} y = -3x + 5 \\ y = (1/3)x - 2/3 \end{cases}\]
I can do this by substituting $y = (1/3)x - 2/3$ into the first equation and solving for $x$.
This gives me $-3x + 5 = (1/3)x - 2/3$.
Multiplying both sides by 3, I get $-9x + 15 = x - 2$.
Adding 9x to both sides, I get $15 = 10x - 2$.
Adding 2 to both sides, I get $17 = 10x$.
Dividing both sides by 10, I get $x = 17/10$.
To find $y$, I plug this value of $x$ into either equation. I'll use the second one.
This gives me $y = (1/3)(17/10) - 2/3$.
Simplifying, I get $y = 17/30 - 20/30 = -1/10$.
Therefore, the point of intersection is $(17/10, -1/10)$.