Problem 32 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line passing through the given point and perpendicular to the given line. $$ (2,6), y=-\frac{1}{2} x+4 $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = 2x + 2\)

Step 1: Find the slope of the new line

The slope of the given line is \(-1/2\). And we know that slope of the line perpendicular to this, which we want to find, is the negative reciprocal of the given line's slope, so we have \(m = - 1 / (-1/2) = 2.\)

Step 2: Use Point-Slope form to get the equation

We start with the point-slope form of the equation which is \(y - y_1 = m(x-x_1)\). In this, \(x_1 = 2\) and \(y_1 = 6\) are the coordinates of the given point, and \(m = 2\) is the slope we calculated in the previous step. Substituting these values we get, \(y - 6 = 2(x - 2)\).

Step 3: Convert Point-Slope form to Slope-intercept form

Simplify the equation from step 2 to get the slope-intercept form: \(y = 2x - 4 + 6, y = 2x +2\). So, the equation of the line perpendicular to the given line and passing through the given point in slope-intercept form is \(y = 2x + 2\).

Key Concepts

Perpendicular Lines

In geometry, perpendicular lines are lines that intersect at a right angle (90 degrees). This distinctive characteristic is what sets them apart from parallel lines, which never meet or intersect. When dealing with the equations of lines on a coordinate plane, the idea of perpendicular lines becomes particularly important.
To find if two lines are perpendicular, we look at their slopes. When lines are perpendicular, their slopes are negative reciprocals of each other. There's a simple yet crucial relationship to remember:

• The product of the slopes of two perpendicular lines is always \(-1\).
• If the slope of one line is \(m\), then the slope of the line perpendicular to it will be \(-1/m\).

These mathematical principles allow us to find the slope of a line that is perpendicular to another given line. It's a foundational concept in algebra that helps us navigate the coordinate system with ease.

Point-Slope Form

The point-slope form is a useful way of writing the equation of a line. It is particularly handy when you know a point the line passes through and its slope. The formula is written as:
\[ y - y_1 = m(x - x_1) \] where:

• \(m\) is the slope of the line,
• \(x_1\) and \(y_1\) are the coordinates of the known point on the line.

This form emphasizes the slope \(m\) and how the line is oriented around a specific point \((x_1, y_1)\). It's a powerful tool because it directly incorporates one of the most vital characteristics of a line, its slope, into the equation.
In practice, once you have identified the slope and the point, you can easily plug these values into the point-slope form formula, and from there, convert it into other forms like the slope-intercept form for further analysis or graphing.

Negative Reciprocal

The concept of negative reciprocals plays a pivotal role in understanding perpendicular lines. When we talk about negative reciprocals in mathematics, we are referring to two numbers whose product is \(-1\). This is crucial when working with slopes of perpendicular lines.
To find the negative reciprocal of a number:

• First, take the reciprocal of the number, which means flipping the numerator and denominator if it's a fraction.
• Then, change the sign of the resulting number to ensure it's negative if it was positive, or positive if it was negative.

For example, the negative reciprocal of \(-1/2\) is \(2\). First, find the reciprocal, which is \(2/1 = 2\), and then apply the negative, turning it from negative to positive. This process underpins why perpendicular slopes, \(-\frac{1}{2}\) and \(2\) in this case, multiply to \(-1\). It's a fundamental idea that ensures lines meet at right angles in a coordinate plane.