The proportional relationship is shown on the graph.Derive the equation of the line y=mx through the origin
To derive the equation of a line in the form y = mx through the origin, we need to determine the value of the slope, m. In a proportional relationship, the slope represents the constant rate of change.
Looking at the graph of the proportional relationship, we can observe that as the x-values increase, the y-values also increase proportionally. This suggests that the slope should be positive.
Since the line passes through the origin (0, 0), we can use any two points on the line to calculate the slope.
Let's choose two points on the line:
Point A: (1, 2)
Point B: (2, 4)
The slope, m, can be calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Using the values from the points A and B:
m = (4 - 2) / (2 - 1)
m = 2 / 1
m = 2
Thus, the slope of the line y = mx through the origin is 2.
The equation of the line can be written as:
y = 2x
To derive the equation of the line y = mx through the origin, we need to determine the value of the slope, m, on the graph.
Since the line passes through the origin, the y-intercept is 0. This means that the equation can be simplified to y = mx.
To find the value of the slope, m, we can select any two points on the graph that lie on the line. Let's say we choose the points (x1, y1) and (x2, y2).
Using the properties of a proportional relationship, we know that the ratio of y2 to x2 is always equal to the ratio of y1 to x1, or:
y2/x2 = y1/x1
Rearranging this equation, we have:
y2/y1 = x2/x1
This implies that the slope of the line, m, is the ratio of the y-coordinates to the x-coordinates of any point on the line.
Therefore, the equation of the line y = mx through the origin can be written as:
y = (y2/x2) * x
or
y = (y1/x1) * x
where (x, y) represents any point on the line.
To derive the equation of a line y = mx through the origin, we need to find the value of the slope, which is represented by "m" in the equation.
In a proportional relationship, the ratio between the y-values and the x-values remains constant. Since the line passes through the origin (0,0), we can use any point on the line to calculate the slope, as all points on this line will have the same slope.
Let's take two points on the line (x1, y1) = (0,0) and (x2, y2) = (1,m) to calculate the slope.
The slope (m) is calculated as:
m = (y2 - y1) / (x2 - x1)
= (m - 0) / (1 - 0)
= m / 1
= m
Since the slope is m itself, we have the equation y = mx through the origin.
So, the equation of the line y = mx represents a proportional relationship, where y is directly proportional to x, with a constant of proportionality equal to the slope, m.