To find the equation of the line passing through the points [tex]\((-4, 2)\)[/tex] and [tex]\((1, -1)\)[/tex] in the standard form [tex]\(Ax + By = C\)[/tex], follow these steps:
1. Calculate the Slope (m):
The formula for 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]
Substituting in the given points [tex]\((-4, 2)\)[/tex] and [tex]\((1, -1)\)[/tex]:
[tex]\[
m = \frac{-1 - 2}{1 + 4} = \frac{-3}{5} = -\frac{3}{5}
\][/tex]
2. Find the y-Intercept (b):
The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
Substitute one of the points to find [tex]\(b\)[/tex]. Using point [tex]\((-4, 2)\)[/tex]:
[tex]\[
2 = -\frac{3}{5}(-4) + b \Rightarrow 2 = \frac{12}{5} + b \Rightarrow b = 2 - \frac{12}{5} = \frac{10}{5} - \frac{12}{5} = -\frac{2}{5}
\][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[
y = -\frac{3}{5}x - \frac{2}{5}
\][/tex]
3. Convert to Standard Form [tex]\(Ax + By = C\)[/tex]:
Rearrange the equation [tex]\(y = -\frac{3}{5}x - \frac{2}{5}\)[/tex]:
[tex]\[
y = -\frac{3}{5}x - \frac{2}{5} \Rightarrow 5y = -3x - 2
\][/tex]
Rearranging gives:
[tex]\[
3x + 5y = -2
\][/tex]
4. Ensure Coefficients are Integers with GCD of 1 and [tex]\(A > 0\)[/tex]:
The equation [tex]\(3x + 5y = -2\)[/tex] is already in the desired form with integer coefficients, and the GCD condition can be confirmed.
- The coefficients are [tex]\(A = 3\)[/tex], [tex]\(B = 5\)[/tex], [tex]\(C = -2\)[/tex].
- The GCD of [tex]\(|3|\)[/tex], [tex]\(|5|\)[/tex], and [tex]\(|-2|\)[/tex] is 1.
- [tex]\(A = 3 > 0\)[/tex], so the condition [tex]\(A > 0\)[/tex] is satisfied.
Thus, the equation of the line in standard form is:
[tex]\[
3x + 5y = -2
\][/tex]
