Question

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

Short answer

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

Step-by-step solution

Identify the coordinates

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

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 \( m = 0/2 = 0 \).

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 \( y = 0 \).

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.