In Exercises 11–38, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = - 3/5, passing through (10, −4)

Verified step by step guidance

1. Identify the point-slope form of a line 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 \( m = -\frac{3}{5} \) and the point \( (x_1, y_1) = (10, -4) \) into the point-slope form equation.
3. The equation becomes: \( y + 4 = -\frac{3}{5}(x - 10) \). This is the point-slope form of the line.
4. To convert to slope-intercept form \( y = mx + b \), distribute the slope on the right side: \( y + 4 = -\frac{3}{5}x + 6 \).
5. Isolate \( y \) by subtracting 4 from both sides to get the slope-intercept form: \( y = -\frac{3}{5}x + 2 \).

Key Concepts

Point-Slope Form

Point-slope form is a way to express the equation of a line when you know a point on the line and its slope. The formula is given by 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 have specific point and slope information.

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 advantageous because it directly shows the slope and where the line crosses the y-axis, making it easy 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. A negative slope, such as -3/5, indicates that as you move to the right along the x-axis, the line descends. Understanding slope is crucial for determining the direction and angle of the line in both point-slope and slope-intercept forms.