Question

the equation of line t is $y = \frac{7}{4}x + 2$. perpendicular to line t is line u, which passes through the point (2, -2). what is the equation of line u? write the equation in slope - intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Find slope of line u

The slope of line \( t \) is \( \frac{7}{4} \). For perpendicular lines, slopes are negative reciprocals. So slope of \( u \) is \( -\frac{4}{7} \).

Step2: Use point - slope form

Point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(2,-2) \) and \( m = -\frac{4}{7} \).
Substitute values: \( y - (-2)=-\frac{4}{7}(x - 2) \)
Simplify: \( y + 2=-\frac{4}{7}x+\frac{8}{7} \)

Step3: Convert to slope - intercept form

Subtract 2 from both sides. \( 2=\frac{14}{7} \), so \( y=-\frac{4}{7}x+\frac{8}{7}-\frac{14}{7} \)
Simplify: \( y = -\frac{4}{7}x-\frac{6}{7} \)

Answer:

\( y = -\frac{4}{7}x-\frac{6}{7} \)