Problem 18 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((-2,0)\) and \((0,2)\)

Step-by-Step Solution

Answer

The point-slope form of the equation of the line passing through points (-2,0) and (0,2) is \(y = x + 2\). And the slope-intercept form of the equation is the same, \(y = x + 2\).

Step 1: Calculate the slope

The slope (m) of the line passing through the points \((-2,0)\) and \((0,2)\) can be calculated by the formula: \(m = (y2 - y1) / (x2 - x1)\). Replace \(x1 = -2\), \(y1 = 0\), \(x2 = 0\) and \(y2 = 2\) into the formula to get \(m = (2 - 0) / (0 - (-2)) = 1\).

Step 2: Substitute into the point-slope form

Now with \(m = 1\), and one of the points on the line \((-2,0)\), use the point-slope equation form: \(y - y1 = m(x - x1)\). Substitute \(x1 = -2\), \(y1 = 0\) and \(m = 1\) to get to the equation \(y - 0 = 1 * (x - (-2))\), which simplifies to \(y = x + 2\).

Step 3: Formulate the slope-intercept form

The equation from step 2 is already simplified to the slope-intercept form \(y = mx + c\), where the slope \(m = 1\) and the y-intercept \(c = 2\). Therefore, the slope-intercept equation for the line is \(y = x + 2\).

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation of a line, which reveals both the slope and the y-intercept of that line. It is written in the form \(y = mx + c\). Here, \(m\) represents the slope of the line, and \(c\) represents the y-intercept, which is the point where the line crosses the y-axis.
To better visualize it:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\), the slope, describes how steep the line is.
• \(c\), the y-intercept, indicates the starting point of the line on the y-axis.

The slope-intercept form is particularly useful when quickly graphing a line, as both the slope and the y-intercept provide straightforward guidance on how to recreate the line on a graph.

How to Calculate the Slope

The slope of a line is a measure of its steepness and direction. Calculating the slope is a crucial first step in understanding the behavior of the line, as it indicates how \(y\) changes with respect to changes in \(x\). You can calculate the slope when given two points on a line by using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
• \(m\) is the slope.

In our given problem, we calculated the slope by substituting the points \((-2,0)\) and \((0,2)\) into the formula:

• \(y_2 - y_1 = 2 - 0 = 2\)
• \(x_2 - x_1 = 0 - (-2) = 2\)
• Thus, \(m = \frac{2}{2} = 1\)

A slope of \(1\) tells us that for each unit increase in \(x\), \(y\) also increases by one unit, indicating a line rising at a 45-degree angle.

Finding the Equation of a Line

To find the equation of a line, it's important to begin with knowing both the slope and one point on the line. Two common ways to express the equation are the point-slope form and the slope-intercept form. Once the slope is known, the point-slope form can be used, expressed as:

\[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. By substituting these into the formula, you can derive the general equation of the line. For instance, using the point \((-2,0)\) and the slope \(1\):

• Formulate: \(y - 0 = 1(x - (-2))\)
• This simplifies to: \(y = x + 2\)

This equation \(y = x + 2\) is now in the slope-intercept form, where \(m = 1\) and \(c = 2\). This concise representation helps easily understand the line's properties and is particularly handy for graphing.