Write an equation in standard form of the line passing through the points [tex]\((2, 6)\)[/tex] and [tex]\((6, 18)\)[/tex].

The equation is [tex]\( y = 3x \)[/tex].

(Type your answer in standard form.)

To convert the equation to standard form [tex]\(Ax + By = C\)[/tex]:

[tex]\[ y = 3x \][/tex]

Subtract [tex]\(3x\)[/tex] from both sides:

[tex]\[ -3x + y = 0 \][/tex]

Thus, the equation in standard form is:

[tex]\[ -3x + y = 0 \][/tex]



Answer :

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]