Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = −1, passing through (−4, − 1/4)

Verified step by step guidance

1. Step 1: Recall the point-slope form of a linear equation, which is given by: $y\; -\; y$₁ = m(x - x₁), where $m$ is the slope and $(x$₁, y₁) is a point on the line.
2. Step 2: Substitute the given slope $m\; =\; -1$ and the point $(x$₁, y₁) = (-4, -1/4) into the point-slope form. This gives: $y\; -\; (-1/4)\; =\; -1(x\; -\; (-4))$.
3. Step 3: Simplify the equation from Step 2. First, simplify the double negatives: $y\; +\; 1/4\; =\; -1(x\; +\; 4)$.
4. Step 4: Expand the equation to convert it into slope-intercept form $y\; =\; mx\; +\; b$. Distribute the slope $-1$ on the right-hand side: $y\; +\; 1/4\; =\; -x\; -\; 4$.
5. Step 5: Isolate $y$ by subtracting $1/4$ from both sides: $y\; =\; -x\; -\; 4\; -\; 1/4$. Combine like terms to simplify the constant term.

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 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 involves rearranging the equation to isolate y.

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 slope of -1 indicates that for every unit increase in x, y decreases by one unit, resulting in a downward slant. Understanding slope is crucial for graphing lines and interpreting their behavior in relation to the coordinate plane.