An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 1 comma 5 right parenthesis and left parenthesis 2 comma 10 right parenthesis.

Question

The graph shows the proportional relationship. Derive the equation of the line y=mx
through the origin.

Answer 1

To derive the equation of the line in the form \( y = mx \), we need to first identify the slope \( m \) of the line. The slope is calculated using two points on the line. In this case, the points given are \( (1, 5) \) and \( (2, 10) \).

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \( (1, 5) \) and \( (2, 10) \):

• \( x_1 = 1 \), \( y_1 = 5 \)
• \( x_2 = 2 \), \( y_2 = 10 \)

Now substituting these values into the slope formula:

\[ m = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 \]

Now that we have found the slope \( m = 5 \), we can write the equation of the line that passes through the origin (0,0) in the proportional relationship form:

\[ y = mx \quad \text{becomes} \quad y = 5x \]

Thus, the equation of the line is:

\[ y = 5x \]