Write an equation in point-slope form and slope-intercept form for the line passing through (-2, -6) and perpendicular to the line whose equation is x − 3y+ 9 = 0.

Verified step by step guidance

1. First, rewrite the given line equation $x-3y+9=0$ in slope-intercept form $y\; =\; mx\; +\; b$ by isolating $y$. Start by moving $x$ and $9$ to the other side: $-3y\; =\; -x\; -\; 9$.
2. Next, divide both sides of the equation by $-3$ to solve for $y$: $y\; =\; \backslash frac\{-x\; -\; 9\}\{-3\}\; =\; \backslash frac\{-x\}\{-3\}\; +\; \backslash frac\{-9\}\{-3\}\; =\; \backslash frac\{1\}\{3\}x\; +\; 3$. So, the slope of the given line is $m\; =\; \backslash frac\{1\}\{3\}$.
3. Since the line we want is perpendicular to the given line, find the negative reciprocal of the slope $\backslash frac\{1\}\{3\}$. The negative reciprocal is $-3$. This will be the slope of the new line.
4. Use the point-slope form of a line equation, which is $y\; -\; y\_1\; =\; m(x\; -\; x\_1)$, where $m$ is the slope and $(x\_1,\; y\_1)$ is the point the line passes through. Substitute $m\; =\; -3$ and the point $(-2,\; -6)$ to get $y\; -\; (-6)\; =\; -3(x\; -\; (-2))$.
5. Simplify the point-slope form equation to get the slope-intercept form. Distribute the slope and simplify: $y\; +\; 6\; =\; -3(x\; +\; 2)$, then $y\; +\; 6\; =\; -3x\; -\; 6$. Finally, subtract 6 from both sides to isolate $y$: $y\; =\; -3x\; -\; 12$.

Key Concepts

Slope of a Line

The slope measures the steepness of a line and is calculated as the ratio of the change in y to the change in x between two points. For a line in standard form Ax + By + C = 0, the slope is -A/B. Understanding slope is essential for finding parallel or perpendicular lines.

Perpendicular Lines and Their Slopes

Two lines are perpendicular if the product of their slopes is -1. This means the slope of a line perpendicular to another with slope m is the negative reciprocal, -1/m. This relationship helps determine the slope of the required line.

Point-Slope and Slope-Intercept Forms of a Line

Point-slope form is written as y - y₁ = m(x - x₁), using a point (x₁, y₁) and slope m. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Both forms are useful for expressing the equation of a line once the slope and a point are known.