Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = -5, passing through (-4, -2)

Verified step by step guidance

1. Start with the point-slope form of a linear equation: $y\; -\; y\_1\; =\; m(x\; -\; x\_1)$, where $m$ is the slope and $(x\_1,\; y\_1)$ is a point on the line.
2. Substitute the given slope $m\; =\; -5$ and the point $(x\_1,\; y\_1)\; =\; (-4,\; -2)$ into the point-slope form. This gives: $y\; -\; (-2)\; =\; -5(x\; -\; (-4))$.
3. Simplify the equation by removing the double negatives: $y\; +\; 2\; =\; -5(x\; +\; 4)$. This is the equation in point-slope form.
4. To convert to slope-intercept form, expand the equation: Distribute $-5$ to both terms inside the parentheses: $y\; +\; 2\; =\; -5x\; -\; 20$.
5. Isolate $y$ by subtracting $2$ from both sides: $y\; =\; -5x\; -\; 22$. This is the equation in slope-intercept form.

Key Concepts

Point-Slope Form

Point-slope form is a way to express the equation of a line when you know the slope and a point on the line. It is written as y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope. This form is particularly useful for quickly writing the equation of a line when given a slope and a specific point.

Slope-Intercept Form

Slope-intercept form is another way to express the equation of a line, defined as y = mx + b, where m is the slope and b is the y-intercept. This form is beneficial for easily identifying the slope and where the line crosses the y-axis. Converting from point-slope to slope-intercept form involves rearranging the equation to isolate y.

Slope

The slope of a line measures its steepness and direction, calculated as the change in y divided by the change in x (rise over run). A negative slope indicates that the line descends from left to right, while a positive slope indicates it ascends. Understanding slope is crucial for graphing lines and interpreting their behavior in relation to other lines.