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 \(2 x+y=2 ;\) containing the point (-3,0)

Short answer

The equation of the line is \( y = \frac{1}{2} x + \frac{3}{2} \).

Step-by-step solution

Identify the slope of the given line

First, rewrite the equation of the line in slope-intercept form, which is \( y = mx + b \). Starting with \( 2x + y = 2 \), solve for \( y \):\( y = -2x + 2 \).The slope \( m \) of this line is \( -2 \).

Find the slope of the perpendicular line

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

Use the point-slope form to find the equation

Using the point-slope form \( y - y_1 = m(x - x_1) \) with the slope \( \frac{1}{2} \) and point \( (-3,0) \), substitute the values:\( y - 0 = \frac{1}{2}(x + 3) \).Simplifying this equation yields:\( y = \frac{1}{2}(x + 3) \).

Convert to the slope-intercept form

Distribute \( \frac{1}{2} \) and simplify:\[ y = \frac{1}{2} x + \frac{3}{2} \].

Key concepts

perpendicular lines

When two lines are perpendicular, they intersect at a right angle (90 degrees). This means the slopes of these lines have a special relationship. Understanding perpendicularity is critical in geometry and algebra, as it helps us find equations of lines relative to each other. For example, if we know the slope of one line, we can easily find the slope of a line perpendicular to it by using the negative reciprocal. This will be explained further in another section.

slope-intercept form

The slope-intercept form of a line’s equation is very user-friendly and is written as: \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. Rewriting an equation in this format makes it much easier to quickly determine these components.
For instance, consider the line given by the equation \(2x + y = 2\). To convert this to slope-intercept form, solve for \(y\): \[ y = -2x + 2 \].
Now it’s clear that:

• The slope \( m \) is \(-2\).
• The y-intercept \( b \) is \(2\).
• Now you can easily graph this line or use these values in further calculations.

point-slope form

The point-slope form is another useful way to write the equation of a line. It looks like this: \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) is a specific point on the line, and \( m \) is the slope.
This form is particularly handy when you know one point on the line and the slope and want to find the full equation.
For example, with a slope of \(\frac{1}{2} \) and passing through the point \((-3, 0)\), substitute these into the point-slope form:

• \( y - 0 = \frac{1}{2}(x + 3) \)
• After simplification, this becomes: \( y = \frac{1}{2}x + \frac{3}{2} \)

This equation represents the line passing through \((-3, 0)\) with the required slope.

negative reciprocal

The term 'negative reciprocal' is key when dealing with perpendicular lines.
If a line has a slope \( m \), the slope of a line perpendicular to it will be \( -\frac{1}{m} \). This ensures that their product is \(-1\), maintaining the perpendicular relationship.
In our example, the original line has a slope of \(-2\). So, the slope of the perpendicular line is:

• \( -\frac{1}{-2} = \frac{1}{2} \)
• This relationship is essential for constructing the equation of the new line, ensuring it intersects the original line at a right angle.

Learning this relationship and how to apply it is crucial for geometry and algebra problems involving perpendicular lines.