Problem 30 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-2,-4) and (1,-1)

Step-by-Step Solution

Answer

Therefore, the slope-intercept form of the line passing through (-2, -4) and (1, -1) is \(y = x - 2\) and its point-slope form is \( y + 1 = x - 1\)

Step 1: Determine the slope

The slope can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Therefore, applying this formula using our points (-2, -4) and (1, -1) we get: \( m = \frac{-1 - (-4)}{1 - (-2)} = 1 \)

Step 2: Use the calculated slope to write the equation in point-slope form

The point-slope form is given by \( y - y_1 = m ( x - x_1) \). Substituting the point (1, -1) and slope \(m = 1 \) into this equation we get: \( y - (-1) = 1 (x - 1) \) or \( y + 1 = x - 1 \)

Step 3: Convert the equation into slope-intercept form

This step involves rearranging the equation found at the end of step 2 in the form: \( y = mx + c) \). Doing so would give us: \( y = x - 2 \).

Key Concepts

Point-Slope Form

The point-slope form is crucial for writing equations of lines when you are given a point on the line and the slope. This is particularly handy when dealing with problems that provide limited information.

The general formula for the point-slope form is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is any point on the line, and \( m \) is the slope of the line. This form highlights the relationship between a general point \( (x, y) \) on the line and the given point by utilizing the consistent slope between any two points on a straight line.

For example, using the coordinates \( (-2, -4) \) and the established slope of 1 from our calculation, the equation in point-slope form is: \[ y - (-4) = 1(x - (-2)) \] Simplified, it becomes: \[ y + 4 = x + 2 \] This equation now gives us the linear relationship in the point-slope format.

Slope-Intercept Form

The slope-intercept form is perhaps one of the most recognized forms of linear equations, represented as: \[ y = mx + c \] In this equation, \( m \) represents the slope, while \( c \) denotes the y-intercept, which is the point where the line crosses the y-axis.

The advantage of this form is the direct read-off of both slope and y-intercept, making it simple to sketch the graph of the line. Converting an equation from point-slope to slope-intercept form involves solving for \( y \) to get the \( y \) alone on one side of the equation.

As demonstrated in the solution above, the point-slope equation \[ y + 1 = x - 1 \] can be rearranged to get the slope-intercept form \[ y = x - 2 \] The equation now includes the slope (1) and the y-intercept (-2), neatly demonstrating how a line with a positive slope intersects the y-axis below the origin.

Slope Calculation

Understanding slope calculation is fundamental in identifying the steepness and direction of a line. The slope, designated by \( m \) in equations, is calculated by determining the rate at which the \( y \) value changes for a corresponding change in the \( x \) value between any two distinct points on a line. The formula to compute the slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two points on the line.

If the slope is positive, the line rises from left to right; if negative, it falls. A zero slope indicates a horizontal line, while an undefined slope signifies a vertical line.

To illustrate, given the points \( (-2, -4) \) and \( (1, -1) \) from our exercise, we plug them into the slope formula to yield: \[ m = \frac{-1 - (-4)}{1 - (-2)} = 1 \] The slope here is 1, signifying a line that rises at a consistent rate; for each step right on the x-axis, it moves an equal step upwards.