Question

write the slope-intercept form of the equation of the line described.

1. through: (4, 0), parallel to $y = -\frac{5}{4}x + 3$
2. through: (1, 5), parallel to $y = 4x$
3. through: (5, -3), perp. to $y = \frac{5}{4}x - 5$
4. through: (-4, -1), perp. to $y = -\frac{2}{3}x - 4$
5. through: (-5, -3), parallel to $y = x + 4$
6. through: (3, 1), parallel to $y = -\frac{4}{3}x - 4$
7. through: (-2, 5), perp. to $y = \frac{2}{5}x - 3$
8. through: (3, -1), perp. to $y = x + 4$

Explanation:

Problem 83:

Step1: Identify slope

Parallel lines have equal slopes. The given line is \( y = -\frac{5}{4}x + 3 \), so slope \( m = -\frac{5}{4} \).

Step2: Use point - slope form

Point - slope form is \( y - y_1 = m(x - x_1) \). We have \( (x_1,y_1)=(4,0) \) and \( m = -\frac{5}{4} \).
Substitute: \( y - 0=-\frac{5}{4}(x - 4) \)

Step3: Simplify to slope - intercept

\( y=-\frac{5}{4}x + 5 \)

Step1: Identify slope

Parallel lines have equal slopes. The given line is \( y = 4x \), so slope \( m = 4 \).

Step2: Use point - slope form

Point - slope form: \( y - y_1=m(x - x_1) \), with \( (x_1,y_1)=(1,5) \) and \( m = 4 \).
Substitute: \( y - 5=4(x - 1) \)

Step3: Simplify to slope - intercept

\( y-5 = 4x-4\), so \( y = 4x + 1 \)

Step1: Identify slope

Perpendicular lines have slopes that are negative reciprocals. The given line is \( y=\frac{5}{4}x - 5 \), so slope of perpendicular line \( m=-\frac{4}{5} \).

Step2: Use point - slope form

Point - slope form: \( y - y_1=m(x - x_1) \), with \( (x_1,y_1)=(5,-3) \) and \( m = -\frac{4}{5} \).
Substitute: \( y+3=-\frac{4}{5}(x - 5) \)

Step3: Simplify to slope - intercept

\( y + 3=-\frac{4}{5}x+4\), so \( y=-\frac{4}{5}x + 1 \)

Answer:

\( y = -\frac{5}{4}x + 5 \)

Problem 84: