Problem 28 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-2,0) and (0,2)

Step-by-Step Solution

Answer

The equation of the line written in point-slope form is \(y = x + 2\) and in slope-intercept form it's \(y = x + 2\). Please note, in this particular case, both forms yield the same result.

Step 1: Calculate the Slope

The slope (m) is calculated by dividing the change in y by the change in x. This can be done using the formula: \(m = (y_2 - y_1) / (x_2 - x_1)\). Plugging in the given points, we find: \(m = (2 - 0) / (0 - -2) = 2 / 2 = 1\). So, the slope of the line is 1.

Step 2: Write in Point-Slope Form

Using the slope calculated in step 1 and one set of coordinates, the equation can be written in point-slope form: \(y - y_1 = m(x - x_1)\). Using the point (-2,0), we get: \(y - 0 = 1*(x - -2)\), which simplifies to: \(y = x + 2\).

Step 3: Write in Slope-Intercept Form

Now the equation will be written in slope-intercept form: \(y = mx + b\). Using the same slope and the point (0,2), we get: \(y = 1*x + 2\), which simplifies to: \(y = x + 2\).

Key Concepts

Slope

The slope of a line is a measure of its steepness or the rate at which it inclines or declines. This is a fundamental concept in algebra that helps us understand how one quantity changes in relation to another. In geometry, the slope is often referred to as "rise over run." This terminology explains how much a line rises vertically for a certain amount of horizontal movement. To calculate the slope (\( m \)), we use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two different points on the line. - A positive slope means the line is rising as it moves from left to right. - A negative slope indicates the line is falling.- If the slope is zero, the line is horizontal, showing no rise or fall. In the given problem, the calculation \( m = \frac{2 - 0}{0 - (-2)} = 1 \), illustrates a positive slope of 1, indicating a perfect diagonal rise.

Point-Slope Form

The point-slope form is a convenient way to express the equation of a line when you know the slope of the line and a single point through which the line passes. This is particularly useful because it's designed to handle situations where the exact y-intercept is unknown, but other key information is on hand.The general formula for the point-slope form is given as:\[y - y_1 = m(x - x_1)\]Here, \( (x_1, y_1) \) are the coordinates of the given point, and \( m \) is the slope.Using the provided point \((-2, 0)\) and the calculated slope \(1\), the point-slope equation becomes:\[y - 0 = 1(x + 2)\]This simplifies to:\(y = x + 2\), showing a straightforward path from the point-slope form to the standard slope-intercept form. This flexibility of form allows for easy manipulation and deeper analysis when graphing linear equations.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express a linear equation, particularly because of its simplicity and directness. This form enables quick identification of both the slope and the y-intercept; essential components for drawing the graph of the line. The slope-intercept equation is expressed as:\[y = mx + b\]where:- \(m\) is the slope.- \(b\) is the y-intercept, the point where the line crosses the y-axis.In the context of our exercise, once the slope was determined to be 1 and using the point (0, 2) as a guide for the y-intercept, the equation directly translates to:\[y = x + 2\]This form quickly communicates that for every unit increase in \(x\), \(y\) increases by the same amount. The y-intercept \(2\) confirms that even when \(x\) is zero, \(y\) starts off at \(2\). This clear and concise form plays a fundamental role in graphing linear relationships quickly and efficiently.