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 \(x-2 y=-5 ;\) containing the point (0,4)

Short answer

\( y = -2x + 4 \)

Step-by-step solution

Identify the slope of the given line

The equation of the given line is: \(x - 2y = -5\). To find the slope, transform this equation into the slope-intercept form \(y = mx + b\).

Convert to slope-intercept form

Rewrite \(x - 2y = -5\) as follows:\[ -2y = -x - 5 \]\[ y = \frac{1}{2}x + \frac{5}{2} \]Here, the slope (m) of the given line is \(\frac{1}{2}\).

Find the perpendicular slope

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \(\frac{1}{2}\) is \(-2\).

Use the point-slope form

With the slope \(-2\) and the point (0,4), use the point-slope form of the equation of a line: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (0, 4)\).

Plug in the values

Substitute \(m = -2\), \(x_1 = 0\), and \(y_1 = 4\) into the point-slope form:\[ y - 4 = -2(x - 0) \]

Simplify the equation

Simplify the equation to get it into slope-intercept form:\[ y - 4 = -2x \]\[ y = -2x + 4 \]

Key concepts

Slope-Intercept Form

To understand the slope-intercept form, start with its general formula: \( y = mx + b \).
Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
For example, in the exercise, the line equation \(x - 2y = -5\) was transformed into slope-intercept form: \( y = \frac{1}{2}x + \frac{5}{2} \).

• First, isolate \( y \) by moving terms involving \( x \) to the other side: \( -2y = -x - 5 \).
• Next, divide everything by −2 to solve for \( y \): \( y = \frac{1}{2}x + \frac{5}{2} \).

So, the slope \( m \) is \( \frac{1}{2} \), and the y-intercept \( b \) is \( \frac{5}{2} \).
This form helps you quickly identify the slope and intercepts, which are crucial for graphing the line and understanding its behavior.

Point-Slope Form

The point-slope form of a line is \( y - y_1 = m (x - x_1) \).
This form uses a known point on the line \((x_1, y_1)\) and the slope \(m\).
In the exercise, after finding the slope of the line perpendicular to \(x - 2y = -5\), which is −2 (more on that in the next section), we use the point (0, 4).

• Plug the slope and the point into the point-slope equation: \( y - 4 = -2 (x - 0) \).
• After simplifying, you get: \( y - 4 = -2x \).

Finally, rearrange this to get the slope-intercept form: \( y = -2x + 4 \).
The point-slope form is versatile and useful when a specific point on the line is known.

Negative Reciprocal

When two lines are perpendicular, their slopes are negative reciprocals of each other.
The negative reciprocal of a number is found by flipping the number and changing the sign.
For instance, if the slope of the first line is \( \frac{1}{2} \), the negative reciprocal is −2.

• Start with the original slope \( m = \frac{1}{2} \).
• Take the reciprocal: \( \frac{1}{2} \rightarrow 2 \).
• Change the sign to get −2.

This concept is crucial in the exercise for finding the slope of the line perpendicular to \(x - 2y = -5\).
Once the negative reciprocal of the original slope is determined, it can be used in the point-slope form to find the equation of the line.