Question

1. the lines represented by the equations $y = \frac{1}{2}x + 4$ and $3x + 6y = 12$ are
2. the same line
3. parallel
4. perpendicular
5. neither parallel nor perpendicular

1. write the equation of the line perpendicular to $y = \frac{1}{2}x + 6$ that passes through the point $(2, -4)$.

write an equation of the line parallel to $y = \frac{1}{2}x + 3$ and through the point $(1, -5)$

Explanation:

Question 4

Step1: Rewrite equations in slope - intercept form

For the first equation \(y+\frac{1}{2}x = 4\), we can rewrite it as \(y=-\frac{1}{2}x + 4\). The slope \(m_1=-\frac{1}{2}\).

For the second equation \(3x + 6y=12\), we solve for \(y\):

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The slope \(m_2 =-\frac{1}{2}\).

Step2: Compare slopes and y - intercepts

Since \(m_1=m_2=-\frac{1}{2}\) and the y - intercepts (\(4\) and \(2\)) are different, the lines are parallel.

Step1: Find the slope of the perpendicular line

The given line is \(y = \frac{1}{2}x+6\). The slope of this line \(m=\frac{1}{2}\). The slope of a line perpendicular to a line with slope \(m\) is \(m_{\perp}=-\frac{1}{m}\). So, \(m_{\perp}=- 2\).

Step2: Use the point - slope form

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

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Step1: Determine the slope of the parallel line

If two lines are parallel, they have the same slope. The slope of the line \(y=\frac{1}{2}x + 3\) is \(m=\frac{1}{2}\), so the slope of the parallel line \(m_{parallel}=\frac{1}{2}\).

Step2: Use the point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(1,-5)\) and \(m = \frac{1}{2}\).

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Answer:

1. parallel

Question 5