Problem 28 - College Algebra

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 equations of the line in two forms are \(y = x + 2\) (Point-Slope form) and \(y = x + 2\) (Slope-Intercept form).

Step 1: Calculate the slope

Given two points (-2,0) and (0,2), the slope of the line can be calculated as \(m = (2-0) / (0 - (-2)) = 1\).

Step 2: Write the Point-Slope Form equation

The Point-Slope form of a line is \(y - y1 = m(x - x1)\). Substituting \(m = 1\), and one of the points \((-2,0)\) we get \(y - 0 = 1 * (x - (-2))\), which simplifies to \(y = x + 2\).

Step 3: Write the Slope-Intercept Form equation

The Slope-Intercept form of a line is \(y = mx + b\). The value of \(m\) is 1 (calculated in Step 1) and the y-intercept \(b\) can be found by substituting the slope \(m\) and a point's coordinates into the equation: \(b = y - mx\). Using the point (0,2): \(b = 2 - 1*0 = 2\). So the equation in slope-intercept form turns out to be \(y = x + 2\).

Key Concepts

point-slope form

The point-slope form is an essential formula in coordinate geometry that helps us to write the equation of a line when we know the slope and one point on the line. This form is very useful for quickly writing an equation without needing to find the y-intercept immediately. It is given by:\[ y - y_1 = m(x - x_1) \]Here, \( m \) is the slope, and \((x_1, y_1)\) is any point on the line. To use this form, you substitute \( m \) and the coordinates of the point into the formula. In our exercise, given the points - \((-2, 0)\)- \((0, 2)\)we calculated the slope as \( 1 \). Using the point \((-2, 0)\), we plug into the formula to get the equation:\[ y - 0 = 1(x - (-2)) \]Simplifying, we arrive at:\[ y = x + 2 \]This step illustrates how flexible the point-slope form is—providing a straightforward way to express a line before converting it to other forms.

slope-intercept form

The slope-intercept form is one of the most popular ways to represent the equation of a line, especially useful for graphing. This is because it directly gives the slope and the y-intercept, making it straightforward to sketch the line. It is represented as:\[ y = mx + b \]where:- \( m \) is the slope- \( b \) is the y-intercept (the point where the line crosses the y-axis)From our solution, with the slope \( m \) already calculated as \( 1 \), we only needed the y-intercept. For this, we used the point \((0, 2)\). Since the x-coordinate for this point is zero, the y-coordinate directly gives us the y-intercept:\[ b = 2 \]Putting it all together, the equation in slope-intercept form for the given line is:\[ y = x + 2 \]The advantage of this form is its clarity in showing both how steep the line is and where it cuts the y-axis, making it highly visual.

calculating slope

Calculating the slope of a line is the first and fundamental step in writing an equation of the line. The slope, often represented by \( m \), measures the steepness or tilt of the line. To calculate the slope when given two points, you use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]These points are usually expressed as - first point \((x_1, y_1)\)- second point \((x_2, y_2)\)For our exercise, with the points \((-2, 0)\) and \((0, 2)\), we substitute into the formula like so:\[ m = \frac{2 - 0}{0 - (-2)} = \frac{2}{2} = 1 \]The result, \( m = 1 \), tells us that the line is rising at an angle where the vertical change (rise) is equal to the horizontal change (run). Understanding slope is key because:

• A positive slope means the line goes upwards as it moves from left to right.
• A negative slope means the line descends.
• A zero slope indicates a perfectly horizontal line.
• An undefined slope represents a vertical line.

Properly calculating the slope not only helps in forming an equation but also gives insight into the line's direction and behavior.