Problem 36 - College Algebra

Question

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

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = -1.5x - 1.5\) and in slope-intercept form it's \(y = -1.5x - 1.5\)

Step 1: Determine the slope (m)

First, calculate the slope of the line, which is the change in \(y\) divided by the change in \(x\) (also known as rise over run) using the given point (1,-3) and the x-intercept (--1, 0). So the slope \(m\) is given by \((0 - (-3)) / ((-1) - 1) = 3 / -2 = -1.5\)

Step 2: Write the equation in point-slope form

Substitute the values of the point (1,-3) and the slope (-1.5) into the point-slope form equation \(y - y_1 = m (x - x_1)\) to get \(y - (-3) = -1.5 (x - 1)\), which simplifies to \(y + 3 = -1.5x + 1.5\). Therefore the equation in point-slope form is \(y = -1.5x + 1.5 - 3\)

Step 3: Write the equation in slope-intercept form

To convert the equation to the slope-intercept form, simplify the right-hand side to get \(y = -1.5x - 1.5\). So the equation of the line in slope-intercept form is \(y = -1.5x - 1.5\)

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line is essential for graphing and analyzing linear relationships easily. It is an equation of a line written as \(y = mx + b\), where \(m\) represents the slope and \(b\) denotes the y-intercept, the point where the line crosses the y-axis. This form provides a straightforward way to visualize the steepness and direction of a line (through the slope) and to identify where the line crosses the y-axis (through the y-intercept).

For instance, in our exercise, after determining the slope and applying it to the point-slope equation, we converted the equation to slope-intercept form: \(y = -1.5x - 1.5\). Here, \(-1.5\) is the slope, indicating that for each step right along the x-axis, the line falls 1.5 units. Additionally, the y-intercept is \(-1.5\), which is the exact location where the line crosses the y-axis.

Equation of a Line

The equation of a line encapsulates all the points that lie along that line. There are several formats for an equation of a line, including point-slope form, slope-intercept form, and standard form. Each provides a different perspective on the line's properties. The point-slope form, \(y - y_1 = m(x - x_1)\), uses a known point \((x_1, y_1)\) and the slope to define the line.

In our provided exercise, after the slope calculation, we inserted the coordinates of the known point and slope into the point-slope form to obtain the equation. It's important to note that no matter the form, the underlying line represented remains unchanged. The conversion from one form to another involves algebraic manipulation but the geometric interpretation, the straight line on a graph, does not alter.

Slope Calculation

The slope of a line measures the line's steepness and direction. To calculate the slope, also known as the 'rise over run,' take any two points on the line and divide the difference in the y-values by the difference in the x-values, represented as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

In our exercise, the slope was calculated using the coordinates of a given point and the x-intercept. It's crucial to apply this formula correctly to find the accurate slope since the slope dictates the tilt of the line on a graph. For the line passing through \((1,-3)\) with an x-intercept of \(-1\), the calculated slope was \(-1.5\), indicating a downward slant from left to right. When students master slope calculation, they gain a powerful tool for analyzing linear trends and relationships in various contexts.