Problem 33 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-3,-1) and (4,-1)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y = -1\) and the slope-intercept form of the line is also \(y = -1\).

Step 1: Calculate the Slope

The slope (m) of a line can be calculated using the formula \(m = (y2 - y1) / (x2 - x1)\). Here, coordinates (x1, y1) = (-3,-1) and (x2, y2) = (4,-1). Putting the values in the formula, we get \(m = (-1 - (-1)) / (4 - (-3)) = 0 / 7 = 0\). So, the slope is 0.

Step 2: Point-Slope Form of the Line

The formula for the point-slope form of the line is \(y - y1 = m(x - x1)\). Substituting m = 0, (x1, y1) = (-3,-1), we get \(y - (-1) = 0*(x - (-3))\) simplifying it, we get \(y + 1 = 0\), or \(y = -1\).

Step 3: Slope-Intercept Form of the Line

The point-slope form of line equation \(y = -1\) already represents the slope-intercept form, as the slope is 0 and the y-intercept is -1. Hence, the slope-intercept form is \(y = mx + c = 0*x - 1 = -1\).

Key Concepts

Point-Slope Form

The point-slope form of a line is a convenient way to write the equation of a line when you know one point on the line and the slope. The formula is:\[y - y_1 = m(x - x_1)\]Here, \( m \) represents the slope, and \((x_1, y_1)\) are the coordinates of the given point. It's like a mathematical template that fills in details to describe a line passing through a given point with a certain inclination.

• To write an equation in point-slope form, you simply need one point on the line and the line’s slope.
• This form is particularly useful when you want to quickly find the equation from two points, especially if slope is easily calculated from them.
• In practice, convert the negative and subtractive signs carefully during substitution to avoid errors.

For the example given, using point \((-3, -1)\) and slope 0, the point-slope form is simply: \[y + 1 = 0(x + 3)\]Which simplifies directly to \( y = -1 \). This equation describes a horizontal line.

Slope-Intercept Form

The slope-intercept form of a line highlights two important aspects of a line: its slope and y-intercept. The general formula for the slope-intercept form is:\[y = mx + c\]In this formula, \( m \) is the slope of the line telling us how steep the line is, while \( c \), also known as the y-intercept, is the point where the line crosses the y-axis.

• The slope-intercept form offers a straightforward view with the slope \( m \) indicating the angle, and \( c \) providing a specific point on the graph.
• It's especially useful for quickly graphing linear equations or analyzing the rise and run of the line.
• When the slope is zero, like in the example, the equation simplifies elegantly since there is no \( x \) term affecting \( y \).

For our task, since the slope is 0, our line remains constant with \( y = -1 \). It serves as an immediate representation of the level, fixed line.

Slope Calculation

Calculating the slope of a line is a fundamental step in understanding line equations. Slope measures the steepness or inclination of a line, and is represented by \( m \). The formula used for slope, derived from two points \((x_1, y_1)\) and \((x_2, y_2)\), is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This calculation gives us the rate at which \( y \) changes as \( x \) changes. Whether positive, negative, or zero, it provides valuable insights into the direction and angle of the line.

• A positive slope means the line inclines upwards from left to right, while a negative slope indicates a downward inclination.
• A slope of zero suggests the line is horizontal, explaining why the \( y \) value remains constant across differing \( x \) values.
• This understanding sets the foundation for converting between different forms of line equations.

For our problem, using points \((-3, -1)\) and \((4, -1)\), we calculate the slope as 0, which signifies a flat, horizontal line on the graph.