Question

given the following coordinates, find the equation of a line in standard form: (12, -15) and (-1, -41).
$2x - y = 39$
$y - 1 = 2(x + 19)$
$y = 2x + 39$
$x - 2y = 39$

Explanation:

Step1: Find the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
For the points \((12, - 15)\) and \((-1, - 41)\), we have \( x_1 = 12,y_1=-15,x_2=-1,y_2 = - 41 \).
\( m=\frac{-41-(-15)}{-1 - 12}=\frac{-41 + 15}{-13}=\frac{-26}{-13}=2 \)

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Let's use the point \((12,-15)\) and \( m = 2 \).
\( y-(-15)=2(x - 12) \)
\( y + 15=2x-24 \)

Step3: Convert to standard form

The standard form of a line is \( Ax+By = C \), where \( A,B,C \) are integers and \( A\geq0 \).
From \( y + 15=2x-24 \), we can rearrange it as \( 2x-y=15 + 24 \)
\( 2x-y=39 \)

We can also check by plugging the points into the equation:
For \((12,-15)\): Left - hand side \(=2\times12-(-15)=24 + 15 = 39\), Right - hand side \( = 39\).
For \((-1,-41)\): Left - hand side \(=2\times(-1)-(-41)=-2 + 41=39\), Right - hand side \( = 39\).

Answer:

A. \( 2x - y=39 \)