Problem 49 - Algebra 1

Question

Write an equation in standard form of the line that passes through the two points. $$(0,0),(2,0)$$

Step-by-Step Solution

Answer

The equation of the line in standard form is \( y = 0 \)

Step 1: Identify the coordinates

The given points are (0,0) and (2,0). Identify these as two points on the line.

Step 2: Calculate the slope of the line

The slope of the line, \( m \), is calculated as the change in y-coordinates divided by the change in x-coordinates. Here, for the two points (0,0) and (2,0), the change in y-coordinates is 0 (since both y-values are 0), and the change in x-coordinates is 2 - 0 = 2. Therefore, the slope of the line \( m = 0/2 = 0 \).

Step 3: Plug into the slope-intercept form of the equation

The slope-intercept form of the equation is \( y = mx + b \). Plug in the slope (\( m=0 \)) and one of the points (0, 0) into this equation to solve for \( b \). Therefore, \( 0 = 0*0 + b \) and this implies that \( b = 0 \). So, the equation of the line is \( y = 0x + 0 \) or simply \( y = 0 \).

Step 4: Convert to standard form

The standard form of a linear equation is \( Ax + By = C \). In this case, the equation of the line in standard form is \( 0x + 1y = 0 \) or simply \( y = 0 \).

Key Concepts

Standard Form of a Line

When talking about the standard form of a line, we refer to a way of expressing the equation of that line in the format of Ax + By = C, where A, B, and C are integers, and A should be a non-negative number. The advantage of the standard form is that it can easily accommodate vertical and horizontal lines which sometimes cannot be represented by other forms.

For example, consider a horizontal line passing through the origin, as in the provided exercise with coordinates (0,0) and (2,0). The slope of this line is zero, which indicates that it is a horizontal line. In standard form, the equation of this horizontal line is written as y = 0. To convert this into standard form, you need to recognize that A (the coefficient of x) can indeed be zero. Thus, representing this equation in standard form would give us 0x + 1y = 0. This is a perfectly valid representation in standard form, as it respects the rule of having A as a non-negative integer (since 0 is neither positive nor negative).

In summary, the standard form is a versatile and structured way to represent the equation of a line, which accounts for all possible line orientations.

Slope Calculation

The slope of a line measures the steepness, incline, or grade of the line, and is typically represented by the symbol m. In a coordinate system, the slope is calculated by the formula m = (y_2 - y_1) / (x_2 - x_1), where (x_1, y_1) and (x_2, y_2) are two distinct points that lie on the line.

In the case of our example with points (0,0) and (2,0), calculating the slope involves finding the difference between the y-coordinates, (0 - 0), and the difference between the x-coordinates, (2 - 0). This gives us a slope of m = 0 / 2, which simplifies to m = 0. A slope of zero means that the line is perfectly horizontal, showing no rise regardless of how much we run along the x-axis. This concept is critical for understanding the behavior of lines in a Cartesian plane and is fundamental in algebra and calculus. The slope is directly connected to the inclination of the line and helps us predict how a line will progress across a graph.

Slope-Intercept Form

The slope-intercept form is another popular way to express the equation of a line, particularly useful when a line is not vertical. This form is written as y = mx + b, where m is the slope of the line, and b is the y-intercept of the line, which is the value of y when x equals zero. This is the point where the line crosses the y-axis.

Using the coordinate points (0,0) and (2,0) from our exercise, we determined that the slope m is zero. Plugging the slope and one of the points into the slope-intercept form, we get y = 0x + b. Since the line runs through the origin, we know that the y-intercept b is also zero, as (0,0) is where the line crosses the y-axis. Hence, the equation simplifies to y = 0. The slope-intercept form is especially intuitive because it directly shows both the inclination of the line and where it intersects the y-axis, making it easy to graph the line from its equation.