To find the equation of the line that passes through the points [tex]\((5, -2)\)[/tex] and [tex]\((-3, 4)\)[/tex], let's go through the process step by step:
1. Compute the slope [tex]\(m\)[/tex] of the line:
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 the given points [tex]\((x_1, y_1) = (5, -2)\)[/tex] and [tex]\((x_2, y_2) = (-3, 4)\)[/tex]:
[tex]\[
m = \frac{4 - (-2)}{-3 - 5} = \frac{4 + 2}{-3 - 5} = \frac{6}{-8} = -\frac{3}{4}
\][/tex]
2. Find the y-intercept [tex]\(b\)[/tex]:
We use the point-slope form of the linear equation [tex]\(y = mx + b\)[/tex] and one of the points to find the y-intercept [tex]\(b\)[/tex]. Using the point [tex]\((5, -2)\)[/tex] as an example:
[tex]\[
-2 = -\frac{3}{4} \cdot 5 + b
\][/tex]
Simplifying and solving for [tex]\(b\)[/tex]:
[tex]\[
-2 = -\frac{15}{4} + b \\
-2 + \frac{15}{4} = b \\
-\frac{8}{4} + \frac{15}{4} = b \\
\frac{7}{4} = b
\][/tex]
3. Write the equation in slope-intercept form:
Now we have the slope [tex]\(m = -\frac{3}{4}\)[/tex] and the y-intercept [tex]\(b = \frac{7}{4}\)[/tex]. The slope-intercept form of the line is:
[tex]\[
y = -\frac{3}{4}x + \frac{7}{4}
\][/tex]
4. Convert to standard form [tex]\(Ax + By + C = 0\)[/tex]:
Multiplying the entire equation by 4 to clear the fractions:
[tex]\[
4y = -3x + 7
\][/tex]
Rearrange to standard form:
[tex]\[
3x + 4y - 7 = 0
\][/tex]
Therefore, the equation of the line passing through the points [tex]\((5, -2)\)[/tex] and [tex]\((-3, 4)\)[/tex] is
[tex]\[
3x + 4y - 7 = 0
\][/tex]
So, the correct answer is:
[tex]\[
3 x + 4 y - 7 = 0
\][/tex]
