Problem 83 - Algebra for College Students

Question

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has \(x\) -intercept \((2,0)\) and \(y\) -intercept \((0,4)\).

Step-by-Step Solution

Answer

The equation of the line in standard form is \( 2x + y = 4 \).

Step 1 - Understand the Intercepts

The given line has an x-intercept at (2, 0) and a y-intercept at (0, 4). Intercepts are points where the line crosses the axes.

Step 2 - Use Intercepts to Find Slope

The slope (m) of a line can be found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, let (x1, y1) = (2, 0) and (x2, y2) = (0, 4): \[ m = \frac{4 - 0}{0 - 2} = \frac{4}{-2} = -2 \]

Step 3 - Use Slope-Intercept Form

The equation of the line in slope-intercept form is: \[ y = mx + b \]From the y-intercept (0, 4), we know that b = 4 (since y = b when x = 0). Therefore, the equation becomes:\[ y = -2x + 4 \]

Step 4 - Convert to Standard Form

The standard form of a line's equation is: \[ Ax + By = C \]Rearrange the equation \( y = -2x + 4 \) to standard form by adding 2x to both sides:\[ 2x + y = 4 \]Thus, the equation in standard form with integral coefficients is:\[ 2x + y = 4 \]

Key Concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. This happens when the y-coordinate is zero. In our problem, the x-intercept is at the point (2, 0). This means that when we plot the line on a graph, it will cross the x-axis at x = 2. Knowing the x-intercept helps in understanding the behavior of the line and is crucial for finding the slope.

y-intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is zero. From the given problem, the y-intercept is at (0, 4). This implies that the line intersects the y-axis at y = 4. The y-intercept is particularly helpful when using the slope-intercept form of the line's equation, as it gives us the value of b in the equation y = mx + b.

slope

The slope of a line describes its steepness and direction. It's calculated as the change in the y-coordinates divided by the change in the x-coordinates between two points on the line. For our line, we use the points (2,0) and (0,4):

\[ m = \frac{4-0}{0-2} = \frac{4}{-2} = -2 \ \]

This tells us the line falls 2 units vertically for every 1 unit it moves horizontally to the right. A negative slope means the line is descending as you move from left to right.

slope-intercept form

The slope-intercept form of a linear equation is given by y = mx + b. Here, m is the slope, and b is the y-intercept. From our calculations:

\[ y = -2x + 4 \ \]

We obtained this equation by using our slope of -2 and the y-intercept of 4. This form is very useful because it directly shows the slope and the y-intercept, making it easier to graph the line.

standard form

The standard form of a linear equation is written as Ax + By = C, where A, B, and C are integers. To convert from the slope-intercept form (y = -2x + 4) to standard form, follow these steps:

1. Add 2x to both sides to get the terms involving x and y on one side: \[ y + 2x = 4 \ \]
2. Rearrange the equation to get it in the standard form: \[ 2x + y = 4 \ \]

This is the equation in standard form with integral coefficients. It's easier to use for some applications, like finding intercepts and solving systems of equations.