Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Perpendicular to the line \(y=\frac{1}{2} x+4 ;\) containing the point (1,-2)

Short answer

y = -2x

Step-by-step solution

Identify the slope of the given line

The given equation is in the slope-intercept form: \(y = \frac{1}{2}x + 4\). The slope (m) of this line is \(\frac{1}{2}\).

Determine the slope of the perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals of each other. Hence, the slope of the perpendicular line is \(-2\) since \(-2 = -\frac{1}{\frac{1}{2}}\).

Write the equation of the line in point-slope form

Use the point-slope form of the equation of a line, which is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the given point. Substituting \(m = -2\) and the point \((1, -2)\), the equation becomes: \(y - (-2) = -2(x - 1)\).

Simplify to slope-intercept form

Simplify the equation from the previous step: \(y + 2 = -2x + 2\). Subtract 2 from both sides to isolate \(y\): \(y = -2x + 0\), which simplifies to \(y = -2x\).

Key concepts

slope-intercept form

The slope-intercept form of a line's equation is one of the most common ways to express the equation of a line. It is written as \[ y = mx + b \] where:

• m is the slope of the line
• b is the y-intercept of the line, which is where the line crosses the y-axis

For example, in the given line equation \[ y = \frac{1}{2}x + 4 \], the slope \( m \) is \( \frac{1}{2} \), and the y-intercept \( b \) is 4. Knowing the slope helps us understand how steep the line is, while the y-intercept tells us the starting point of the line on the y-axis.

negative reciprocal

When two lines are perpendicular, their slopes are negative reciprocals of each other. This means that if the slope of one line is \( m \), the slope of the other line will be \( -\frac{1}{m} \). For example, in the exercise, the slope of the given line is \( \frac{1}{2} \), and therefore, the slope of the line perpendicular to it will be \( -2 \), calculated as follows:

• First, find the reciprocal of \( \frac{1}{2} \), which is \( 2 \)
• Then, change the sign to get \( -2 \)

This new slope \( -2 \) is crucial to writing the equation of our perpendicular line.

point-slope form

The point-slope form is another useful way to express the equation of a line, particularly when you know a point on the line and the slope. It is written as \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope and \( (x_1, y_1) \) is a specific point on the line. In the exercise, we're given the point \( (1, -2) \) and the slope \( -2 \). Plugging these values into the point-slope form gives us: \[ y - (-2) = -2(x - 1) \] Simplifying this, we get:

• \[ y + 2 = -2(x - 1) \]
• Distribute the \( -2 \) across \( (x - 1) \) to get \[ y + 2 = -2x + 2 \]
• Finally, isolate \( y \) by subtracting 2 from both sides to get the slope-intercept form: \[ y = -2x \]

Now we have our equation in a clean, understandable form.