The proportional relationship is shown on the graph. Derive the equation of the line y=mx through the origin.
In a proportional relationship, the equation of the line through the origin is given by y = mx, where m is the slope of the line.
To derive the equation of the line y = mx from the graph, we need to find the slope (m) of the line.
The slope of a line can be found by taking any two points on the line and using the formula: m = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
In this case, since the line passes through the origin, the y-intercept is 0, and any point on the line can be used to find the slope.
Let's take two points on the line from the graph. Let's say (x1, y1) = (2, 4) and (x2, y2) = (4, 8).
Using the formula for slope, we have:
m = (y2 - y1)/(x2 - x1)
= (8 - 4)/(4 - 2)
= 4/2
= 2
So the slope (m) of the line is 2.
Therefore, the equation of the line y = mx through the origin is y = 2x.
To derive the equation of the line y = mx through the origin, we need to determine the value of m, the slope of the line.
Since the line passes through the origin, which is (0, 0), we can use any other point on the line to find the slope.
Let's consider a point (x, y) that lies on the line. Since it's a proportional relationship, the ratio of y to x should be constant. Therefore, we have:
y / x = k
where k is the constant of proportionality.
Since we know that y = mx (from the equation of the line), we can substitute this into the equation above:
mx / x = k
This simplifies to:
m = k
So, the slope of the line y = mx through the origin is equal to k.
Hence, the equation of the line y = mx through the origin is:
y = mx
To derive the equation of the line y = mx through the origin, you need to determine the slope of the line.
In a proportional relationship, the ratio of the y-coordinates to the x-coordinates remains constant. This constant ratio is the slope of the line.
To determine the slope from the graph, you can choose any two points on the line. Since the line passes through the origin (0, 0), you can choose another point on the line.
Let's take another point on the line with coordinates (x1, y1).
Using the slope formula, the slope (m) can be calculated as:
m = (y1 - 0) / (x1 - 0) = y1 / x1
Now that you know the slope (m) of the line, you can write the equation y = mx through the origin using the slope-intercept form of a line equation:
y = mx
Therefore, the equation of the line through the origin is y = mx, where m is the slope of the line determined from the graph.