Question

1. a line that passes through the points (-1, -4) and (4, 2) is shown. which equation is the equation of the line that is perpendicular to the given line and passes through the point (-3, 7)? -6x + 5y = 53 -3x + 2y = 23 2x + 3y = 27 5x + 6y = 27

Explanation:

Step1: Find slope of given line

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For \((-1, -4)\) and \((4, 2)\), \( m=\frac{2 - (-4)}{4 - (-1)}=\frac{6}{5} \).

Step2: Find slope of perpendicular line

The slope of a line perpendicular to a line with slope \( m \) is \( -\frac{1}{m} \). So, the slope of the perpendicular line is \( -\frac{5}{6} \).

Step3: Use point - slope form

The point - slope form is \( y - y_1=m(x - x_1) \). Using the point \((-3, 7)\) and slope \( -\frac{5}{6} \), we have \( y - 7=-\frac{5}{6}(x + 3) \).
Multiply through by 6 to eliminate the fraction: \( 6y-42=-5x - 15 \).
Rearrange to standard form: \( 5x+6y=27 \).

Answer:

\( 5x + 6y = 27 \) (the fourth option: 5x + 6y = 27)