Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. \(x\) -intercept \(=2 ; y\) -intercept \(=-1\)

Short answer

The equation in slope-intercept form is \( y = \frac{1}{2}x - 1 \), and in general form, it is \( x - 2y - 2 = 0 \).

Step-by-step solution

Understanding Intercepts

The problem gives us the x-intercept at 2, which translates to the point (2, 0), and the y-intercept at -1, corresponding to the point (0, -1). These two points will help us determine the equation of the line.

Find the Slope

The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \). Substituting in our points (2, 0) and (0, -1), the slope \( m = \frac{(0 - (-1))}{(2 - 0)} = \frac{1}{2} \).

Use the Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b\), where \( m\) is the slope and \( b\) is the y-intercept. We already know the slope \( m = 1/2\) and the y-intercept \( b = -1\). Therefore, the equation of the line is \( y = \frac{1}{2}x - 1 \).

Convert to General Form (Optional)

To express the equation in the general form \( Ax + By + C = 0 \), we start with \( y = \frac{1}{2}x - 1 \). Multiply through by 2 to clear the fraction: \( 2y = x - 2 \), then rearrange to get \( -x + 2y + 2 = 0 \). Alternatively, write it as \( x - 2y - 2 = 0 \).

Key concepts

slope-intercept form

The slope-intercept form of the equation of a line is one of the most commonly used formats. The general formula for this form is \(y = mx + b\), where:

• \(m\) represents the slope of the line
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis

For example, if you have a slope of \( \frac{1}{2} \) and a y-intercept of \( -1 \), plugging these values into the formula gives you the equation \( y = \frac{1}{2} x - 1 \). This form is great because it immediately tells you the slope and the starting point of the line.
Understanding this form makes it very easy to graph a line since you can start at the y-intercept and use the slope to find other points.

general form

The general form of the equation of a line is another way to express linear equations. It is written as \( Ax + By + C = 0 \). Here:

• \(A, B,\) and \(C\) are constants
• \(x\) and \(y\) are variables representing the coordinates on the Cartesian plane

To convert from slope-intercept form to general form, you can rearrange the equation \( y = \frac{1}{2} x - 1 \) to eliminate the fraction and move all terms to one side of the equation:
1. Multiply through by 2: \( 2y = x - 2 \)
2. Rearrange: \( -x + 2y + 2 = 0 \) or simply \( x - 2y - 2 = 0 \)
This form can be useful in certain types of mathematical problems and engineering applications.

intercepts

Intercepts are points where the line crosses the axes. There are two kinds of intercepts to consider:

• x-intercept: Where the line crosses the x-axis. At this point, \( y = 0 \).
• y-intercept: Where the line crosses the y-axis. At this point, \( x = 0 \).

In the given problem:
The x-intercept is 2 which means the point is (2,0).
The y-intercept is -1, thus the point is (0,-1).
You can determine these intercepts just by setting the other variable to zero in the equation of the line.

slope calculation

The slope of a line is a measure of its steepness and is usually denoted by \( m \). It can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two points on the line. Using the provided intercepts, (2, 0) and (0, -1), the slope \( m \) is calculated as:
\[ m = \frac{0 - (-1)}{2 - 0} = \frac{1}{2} \]
This slope tells us that for every 2 units you move horizontally to the right, the line will move up by 1 unit. The slope is essential for writing an equation of the line in slope-intercept form.