Question

line t has an equation of $y = \frac{4}{3}x - 2$. line u is perpendicular to line t and passes thr (-2, 2). what is the equation of line u? write the equation in slope-intercept form. write the numbers in the equation as sim proper fractions, improper fractions, or integers.

Explanation:

Step1: Find slope of line u

The slope of line \( t \) is \( \frac{4}{3} \). For perpendicular lines, the slope of line \( u \) (\( m_u \)) is the negative reciprocal, so \( m_u = -\frac{3}{4} \).

Step2: Use point - slope form

The point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(-2,2) \) and \( m = -\frac{3}{4} \).
Substitute values: \( y - 2 = -\frac{3}{4}(x - (-2)) \)
Simplify: \( y - 2 = -\frac{3}{4}(x + 2) \)

Step3: Convert to slope - intercept form

Expand the right - hand side: \( y - 2 = -\frac{3}{4}x-\frac{3}{2} \)
Add 2 to both sides: \( y = -\frac{3}{4}x-\frac{3}{2}+2 \)
Simplify the constant terms: \( y = -\frac{3}{4}x+\frac{1}{2} \)

Answer:

\( y = -\frac{3}{4}x+\frac{1}{2} \)