To find the equation of a line in standard form [tex]\(Ax + By = C\)[/tex] that passes through the points [tex]\((2, 6)\)[/tex] and [tex]\((6, 18)\)[/tex], follow these steps:
1. Calculate the slope [tex]\(m\)[/tex] of the line:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((2, 6)\)[/tex] and [tex]\((6, 18)\)[/tex]:
[tex]\[
m = \frac{18 - 6}{6 - 2} = \frac{12}{4} = 3
\][/tex]
2. Use the point-slope form of the equation:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the slope [tex]\(m = 3\)[/tex] and the point [tex]\((2, 6)\)[/tex]:
[tex]\[
y - 6 = 3(x - 2)
\][/tex]
Simplify this equation:
[tex]\[
y - 6 = 3x - 6
\][/tex]
Adding 6 to both sides to get it into slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = 3x
\][/tex]
3. Convert the slope-intercept form [tex]\(y = 3x\)[/tex] to standard form:
The standard form of a linear equation is [tex]\(Ax + By = C\)[/tex]. To convert [tex]\(y = 3x\)[/tex] into this form, rewrite it as:
[tex]\[
3x - y = 0
\][/tex]
Therefore, the equation in standard form of the line passing through the points [tex]\((2, 6)\)[/tex] and [tex]\((6, 18)\)[/tex] is:
[tex]\[
3x - y = 0
\][/tex]
