Question

1. write the equation of the line perpendicular to y = ½ x + 3 that passes through the point (2, 4).

write an equation of the line parallel to y = ½ x + 3 and through the point (1, -5)

Explanation:

First, let's solve the problem of writing the equation of the line parallel to \( y = \frac{1}{2}x + 3 \) and through the point \( (1, -5) \)

Step 1: Recall the slope of parallel lines

Parallel lines have the same slope. The given line \( y = \frac{1}{2}x + 3 \) is in slope - intercept form \( y=mx + b \), where \( m \) is the slope. So the slope of the given line \( m=\frac{1}{2} \), and the slope of the line we want to find (let's call it \( m_{parallel} \)) is also \( \frac{1}{2} \).

Step 2: Use the point - slope form of a line

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 \( m=\frac{1}{2} \) and the point \( (x_1,y_1)=(1, - 5) \). Substitute these values into the point - slope form:
\( y-(-5)=\frac{1}{2}(x - 1) \)

Step 3: Simplify the equation to slope - intercept form

Simplify the left - hand side: \( y + 5=\frac{1}{2}(x - 1) \)
Distribute the \( \frac{1}{2} \) on the right - hand side: \( y+5=\frac{1}{2}x-\frac{1}{2} \)
Subtract 5 from both sides to solve for \( y \). We know that \( 5=\frac{10}{2} \), so:
\( y=\frac{1}{2}x-\frac{1}{2}-\frac{10}{2} \)
\( y=\frac{1}{2}x-\frac{11}{2} \)

Now, let's solve the problem of writing the equation of the line perpendicular to \( y=\frac{1}{2}x + 3 \) (assuming the original first problem has the given line \( y = \frac{1}{2}x+3 \)) and through the point \( (2,4) \)

Step 1: Recall the slope of perpendicular lines

If two lines are perpendicular, the product of their slopes is \( - 1 \). Let the slope of the given line be \( m_1=\frac{1}{2} \) and the slope of the perpendicular line be \( m_2 \). Then \( m_1\times m_2=-1 \). So \( \frac{1}{2}\times m_2=-1 \), which gives \( m_2=-2 \).

Step 2: Use the point - slope form of a line

The point - slope form is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(2,4) \) and \( m = - 2 \). Substitute these values into the formula:
\( y - 4=-2(x - 2) \)

Step 3: Simplify the equation to slope - intercept form

Distribute the \( - 2 \) on the right - hand side: \( y - 4=-2x + 4 \)
Add 4 to both sides: \( y=-2x+4 + 4 \)
\( y=-2x + 8 \)

For the first sub - question (parallel line):

Answer:

\( y=\frac{1}{2}x-\frac{11}{2} \) (or \( y = 0.5x-5.5 \))

For the second sub - question (perpendicular line):