Question

practice exercises
in exercises 1–4, write an equation for line l in point - slope form and slope - intercept form.
1.
l is parallel to ( y = 2x ).
2.
l is parallel to ( y = - 2x ).
3.
l is perpendicular to ( y = 2x ).
4.
l is perpendicular to ( y = - 2x ).

Explanation:

Let's solve Exercise 1 first.

Step 1: Recall the properties of parallel lines

Parallel lines have the same slope. The given line is \( y = 2x \), so its slope \( m = 2 \). Thus, line \( L \), being parallel to \( y = 2x \), also has a slope \( m = 2 \).

Step 2: Use the point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1) \) is a point on the line and \( m \) is the slope. We know that the line \( L \) passes through the point \( (4,2) \) and has a slope \( m = 2 \). Substituting \( x_1 = 4 \), \( y_1=2 \) and \( m = 2 \) into the point - slope form, we get:
\( y - 2=2(x - 4) \)

Step 3: Convert to slope - intercept form (\( y=mx + b \))

We start with the point - slope form \( y - 2=2(x - 4) \).
First, distribute the 2 on the right - hand side: \( y-2 = 2x-8 \).
Then, add 2 to both sides of the equation: \( y=2x-8 + 2 \), which simplifies to \( y = 2x-6 \).

Answer:

(for Exercise 1)

• Point - slope form: \( y - 2=2(x - 4) \)
• Slope - intercept form: \( y = 2x-6 \)

Now, let's solve Exercise 2.

Step 1: Determine the slope of line \( L \)

Parallel lines have equal slopes. The line \( y=-2x \) has a slope \( m=-2 \). Since line \( L \) is parallel to \( y = - 2x \), the slope of line \( L \), \( m=-2 \).

Step 2: Use the point - slope form

The line \( L \) passes through the point \( (3,4) \). Using the point - slope form \( y - y_1=m(x - x_1) \) with \( x_1 = 3 \), \( y_1 = 4 \) and \( m=-2 \), we have:
\( y - 4=-2(x - 3) \)

Step 3: Convert to slope - intercept form

Start with \( y - 4=-2(x - 3) \).
Distribute the - 2: \( y-4=-2x + 6 \).
Add 4 to both sides: \( y=-2x+6 + 4 \), so \( y=-2x + 10 \).