Problem 38 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. \(x\) -intercept \(-4\) and \(y\) -intercept \(--2\)

Step-by-Step Solution

Answer

The line's equation in the point-slope form is \(y - 0.5x + 2 = 0\) and in the slope-intercept form is \(y = -0.5x - 2\).

Step 1: Find the Slope

Since the two points given are the intercepts, they are \((-4, 0)\) for the x-intercept and \((0, -2)\) for the y-intercept. The formula to find the slope \(m\) is \((y_2 - y_1) / (x_2 - x_1) = (-2 - 0)/(0 - -4) = -2/4 = -0.5.

Step 2: Write the Point-slope Form

The general point-slope formula is \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Let's use the y-intercept \((0, -2)\) as the point. Plugging the values, we get \(y - (-2) = -0.5(x - 0)\), simplifying it gives \(y + 2 = -0.5x\). Hence, \(y - 0.5x + 2 = 0\) is the point-slope form of the equation.

Step 3: Write the Slope-Intercept Form

The general slope-intercept form is \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. We already know from step 1 that the slope \(m\) is -0.5 and y-intercept \(b\) is -2. Substitute these values to the slope-intercept formula, we get \(y = -0.5x - 2\).

Key Concepts

Point-Slope Form

The point-slope form of an equation is a convenient way to describe a line when you know the slope and one point on the line. This form is expressed as:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) is a known point on the line, and \(m\) is the line's slope. To use this form, simply plug in the values for the slope and the chosen point's coordinates. In our example, the y-intercept \((0, -2)\) was used as the point, and the slope \(-0.5\). This results in the point-slope equation:

• \(y + 2 = -0.5(x - 0)\)

This formula is great for quickly shifting a line's representation from just a slope and one point to a usable algebraic expression.

Slope-Intercept Form

The slope-intercept form is perhaps the most recognizable way to represent a linear equation. This form is written as:

• \(y = mx + b\)

In this formula, \(m\) stands for the slope of the line, while \(b\) represents the y-intercept—the point where the line crosses the y-axis. This form makes it extremely simple to understand a line's steepness and where it will intersect the y-axis. In our solved exercise, we identified the slope \(-0.5\) and the y-intercept \(-2\) to form the slope-intercept equation:

• \(y = -0.5x - 2\)

By using this form, you can quickly graph lines and understand their behavior relative to the axes.

Intercepts

Intercepts are key points where a line crosses the x or y-axis. Knowing these points can facilitate graphing the line without having to solve for additional points.

• X-intercept: The point where the line crosses the x-axis (y=0).
• Y-intercept: The point where the line crosses the y-axis (x=0).

In the problem at hand, the x-intercept is \((-4, 0)\) and the y-intercept is \((0, -2)\). These two intercepts are especially useful because they let you determine the slope and quickly develop equations for the line in both point-slope and slope-intercept forms. Embracing intercepts is beneficial in visualizing and plotting linear equations.

Slope Calculation

Calculating the slope of a line is a fundamental task when writing equations. The slope, often represented as \(m\), tells us how steep the line is. It is calculated using the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula relies on choosing two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\). The difference in the y-values is divided by the difference in the x-values. For the given intercepts (\((-4, 0)\) and \((0, -2)\)), the slope is calculated as follows:

• \(m = \frac{-2 - 0}{0 - (-4)} = \frac{-2}{4} = -0.5\)

Accurately calculating the slope is crucial for creating both the point-slope and slope-intercept equations.