Question

In Exercises \(31-34,\) write an equation for the line through \(P\) that is (a) parallel to \(L,\) and (b) perpendicular to \(L .\) $$P(-2,2), \quad L : 2 x+y=4$$

Short answer

The equation of the line parallel to \(L\) through \(P\) is \(y = -2x - 2\). The equation of the line perpendicular to \(L\) through \(P\) is \(y = 1/2x + 3\).

Step-by-step solution

Find the slope of line \(L\)

The equation of \(L\) is in the form of \(ax + by = c\). To convert it into the slope-intercept form \(y = mx + b\), where \(m\) represents the slope, we can rearrange the equation: \(y = -2x + 4\). So, the slope of \(L\) is -2.

Write the equation of the line parallel to \(L\)

Since parallel lines have the same slope, the slope of the line parallel to \(L\) passing through \(P(-2,2)\) is also -2. Thus, the equation of the line will be in the form \(y - y1 = m(x - x1)\), where \((x1, y1)\) are the coordinates of \(P\). Substituting for \(P, m\) and simplifying, we get: \(y - 2 = -2 (x + 2)\) or \(y = -2x - 2\).

Write the equation of the line perpendicular to \(L\)

Slopes of perpendicular lines are negative reciprocals. Therefore, the slope of the required line is \(1/2\). Again, we will use the standard formula \(y - y1 = m(x - x1)\). Substituting for \(P, m\) and simplifying, we get the equation: \(y - 2 = 1/2 (x + 2)\) or \(y = 1/2x + 3\).

Key concepts

Parallel Lines

Parallel lines are lines that run alongside each other and never intersect, regardless of how far they are extended. A noteworthy property of parallel lines is that they have the same slope. When you are asked to write an equation for a line parallel to a given line, like with line L : 2x + y = 4, the first step is always to determine the slope of the original line. In this case, by rearranging the equation into the slope-intercept form y = mx + b, we find the slope (m) to be -2. For the line to be parallel through point P(-2,2), we use the same slope of -2 and apply it to the point-slope formula, resulting in the equation y = -2x - 2. This line shares the same slope as line L but passes through the specific point P.

Perpendicular Lines

Perpendicular lines are lines that intersect at right angles (90 degrees) with each other. The relationship between the slopes of two perpendicular lines is that they are negative reciprocals of each other. This means if the slope of one line is a, the slope of a line perpendicular to it will be -1/a. After finding the slope of line L to be -2, the slope of the perpendicular line will be the negative reciprocal of -2, which is 1/2. To write the equation of a line perpendicular to L that goes through point P(-2,2), we use the slope 1/2 and the point-slope form, yielding y = 1/2x + 3.

Slope-Intercept Form

The slope-intercept form of a line's equation is one of the most straightforward expressions used in algebra for a linear equation, represented as y = mx + b. In this formula, m stands for the slope, while b is the y-intercept, which is the point where the line crosses the y-axis. Using the slope-intercept form makes it easier to graph the line and understand its properties. It's also beneficial for comparing the slopes of parallel and perpendicular lines, as seen with line L which was rearranged to y = -2x + 4, indicating a slope of -2.

Point-Slope Form

When writing equations of lines, another useful form is the point-slope form, especially when you have a point through which the line passes and its slope. The formula is y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of the given point. In our case, we used this formula to write the equations of the lines that are parallel and perpendicular to the original line L. For the parallel line, we substituted point P(-2,2) and the slope -2 into the form. Similarly, for the perpendicular line, we used the same point P with the slope of 1/2, giving us the respective equations after simplification.

Negative Reciprocal

Understanding the concept of a negative reciprocal is crucial when working with perpendicular lines in coordinate geometry. If two lines are perpendicular, the slope of one line is the negative reciprocal of the other. In mathematical terms, if one line has a slope of m, the perpendicular line's slope will be -1/m. The term reciprocal refers to flipping a fraction (or considering 1 divided by the number if it's an integer), and negative implies changing the sign. This relationship ensures that the resulting lines cross at 90 degrees. By using the negative reciprocal, we were able to find that the slope of the line perpendicular to line L (with a slope of -2) is 1/2.