IDNLearn.com is your go-to resource for finding expert answers and community support. Discover prompt and accurate responses from our experts, ensuring you get the information you need quickly.

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]
Your engagement is important to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.