In Exercises 41–44, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-3, 2) with slope - 6

Verified step by step guidance

1. Identify the point-slope form of a linear equation, which is given by: $y\; -\; y\_1\; =\; m(x\; -\; x\_1)$, where $m$ is the slope and $(x\_1,\; y\_1)$ is a point on the line.
2. Substitute the given slope $m\; =\; -6$ and the point $(x\_1,\; y\_1)\; =\; (-3,\; 2)$ into the point-slope form. This gives: $y\; -\; 2\; =\; -6(x\; -\; (-3))$.
3. Simplify the equation from step 2 to get the point-slope form: $y\; -\; 2\; =\; -6(x\; +\; 3)$.
4. To convert to slope-intercept form, expand the equation from step 3. Distribute $-6$ across $(x\; +\; 3)$, resulting in: $y\; -\; 2\; =\; -6x\; -\; 18$.
5. Solve for $y$ by adding $2$ to both sides of the equation: $y\; =\; -6x\; -\; 16$. This is the slope-intercept form of the equation.

Key Concepts

Point-Slope Form

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

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 allows for a clearer understanding of the line's behavior.

Slope

The slope of a line measures its steepness and direction, calculated as the change in y over the change in x (rise over run). A positive slope indicates the line rises as it moves from left to right, while a negative slope indicates it falls. In this question, the slope is given as -6, meaning for every unit increase in x, y decreases by 6 units, which is crucial for forming the line's equation.