Find answers to your questions and expand your knowledge with IDNLearn.com. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.

Write the equation of the line passing through the points [tex]\((-4, 2)\)[/tex] and [tex]\((1, -1)\)[/tex] in the form [tex]\(Ax + By = C\)[/tex], where [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] are integers with greatest common divisor 1, and [tex]\(A\)[/tex] is positive.

Sagot :

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]