To determine which equation represents the line passing through the points [tex]\((2, 3)\)[/tex] and [tex]\((4, 7)\)[/tex], let's follow these steps:
1. Calculate the Slope (m):
The slope [tex]\( m \)[/tex] of a line passing 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]
Here, substituting in the given points [tex]\((x_1, y_1) = (2, 3)\)[/tex] and [tex]\((x_2, y_2) = (4, 7)\)[/tex]:
[tex]\[
m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2
\][/tex]
2. Point-Slope Form:
Using the slope [tex]\( m \)[/tex] and one of the points, e.g., [tex]\((2, 3)\)[/tex], we can write the equation of the line in point-slope form, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\( m = 2 \)[/tex] and the point [tex]\((2, 3)\)[/tex]:
[tex]\[
y - 3 = 2(x - 2)
\][/tex]
3. Convert to Slope-Intercept Form:
Rearranging this equation into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 3 = 2x - 4
\][/tex]
[tex]\[
y = 2x - 1
\][/tex]
4. Convert to Standard Form:
To convert the slope-intercept form to the standard form [tex]\( Ax + By = C \)[/tex]:
[tex]\[
y = 2x - 1 \implies 2x - y = 1
\][/tex]
Therefore, the standard form of the equation representing the line passing through the points [tex]\((2, 3)\)[/tex] and [tex]\((4, 7)\)[/tex] is:
[tex]\[
\boxed{2x - y = 1}
\][/tex]
From the options provided, the correct answer is:
[tex]\[
2x - y = 1
\][/tex]
