Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = 8, passing through (4, −1)

Verified step by step guidance

1. Step 1: Recall the point-slope form of a linear equation, which is given by: $y\; -\; y\_1\; =\; m(x\; -\; x\_1)$, where $m$ is the slope and $(x\_1,\; y\_1)$ is a point on the line.
2. Step 2: Substitute the given slope $m\; =\; 8$ and the point $(x\_1,\; y\_1)\; =\; (4,\; -1)$ into the point-slope form. This gives: $y\; -\; (-1)\; =\; 8(x\; -\; 4)$.
3. Step 3: Simplify the equation from Step 2. The double negative becomes positive, so the equation becomes: $y\; +\; 1\; =\; 8(x\; -\; 4)$. This is the equation in point-slope form.
4. Step 4: To convert to slope-intercept form, expand the equation $y\; +\; 1\; =\; 8(x\; -\; 4)$. Distribute the $8$ to get: $y\; +\; 1\; =\; 8x\; -\; 32$.
5. Step 5: Isolate $y$ by subtracting $1$ from both sides of the equation. This gives: $y\; =\; 8x\; -\; 33$. This is the equation in slope-intercept form.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is expressed 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 writing equations when you know a point on the line and the slope, allowing for straightforward calculations and graphing.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope and b is the y-intercept. This form is advantageous for quickly identifying the slope and the point where the line crosses the y-axis, making it easier to graph the line and understand its behavior.

Slope

Slope is a measure of the steepness or incline of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this case, a slope of 8 indicates that for every unit increase in x, y increases by 8 units, which significantly influences the line's angle and position.