To determine the equation of the line passing through the points [tex]\((5,2)\)[/tex], [tex]\((10,4)\)[/tex], and [tex]\((15,6)\)[/tex], we need to follow a few steps carefully:
1. Calculate the Slope (m):
The slope of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\((5,2)\)[/tex] and [tex]\((10,4)\)[/tex]:
[tex]\[ m = \frac{4 - 2}{10 - 5} = \frac{2}{5} \][/tex]
Thus, the slope [tex]\( m \)[/tex] is [tex]\( 0.4 \)[/tex].
2. Determine the Y-intercept (b):
Once we have the slope, we use the line equation in the slope-intercept form:
[tex]\[ y = mx + b \][/tex]
We can substitute one of the points and the slope to find [tex]\( b \)[/tex]. Using the point [tex]\((5, 2)\)[/tex] and [tex]\( m = 0.4 \)[/tex]:
[tex]\[ 2 = 0.4 \cdot 5 + b \][/tex]
[tex]\[ 2 = 2 + b \][/tex]
[tex]\[ b = 0 \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex].
3. Formulate the Line Equation:
With [tex]\( m = 0.4 \)[/tex] and [tex]\( b = 0 \)[/tex], the equation of the line is:
[tex]\[ y = 0.4x \][/tex]
Thus, the correct equation of the line that passes through the points [tex]\((5, 2)\)[/tex], [tex]\((10, 4)\)[/tex], and [tex]\((15, 6)\)[/tex] is:
[tex]\[ y = \frac{2}{5} x \][/tex]
Therefore, the correct option is:
A. [tex]\( y = \frac{2}{5} x \)[/tex]
