Problem 51 - A Graphical Approach to College Algebra

Question

Find the equation of the line satisfying the given conditions, giving it in slope-intercept form if possible. Through the origin, perpendicular to \(2 x+y=6\)

Step-by-Step Solution

Answer

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

Step 1: Identify and Rewrite Given Equation in Slope-Intercept Form

The given equation is \(2x + y = 6\). To find its slope, first rewrite it in slope-intercept form (\(y = mx + b\)) by isolating \(y\):\[ y = -2x + 6 \] Here, the slope \(m\) is \(-2\).

Step 2: Determine the Perpendicular Slope

Lines that are perpendicular to each other have slopes that are negative reciprocals. Since the slope of the given line is \(-2\), the perpendicular slope \(m\) is \(\frac{1}{2}\) (the negative reciprocal of \(-2\) is \(\frac{1}{2}\)).

Step 3: Use Point-Slope Form to Find the Equation

The perpendicular line passes through the origin \((0, 0)\) and has a slope of \(\frac{1}{2}\). Use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), with point \((x_1, y_1) = (0, 0)\) and slope \(m = \frac{1}{2}\):\[ y - 0 = \frac{1}{2}(x - 0) \] Simplify to obtain: \[ y = \frac{1}{2}x \]

Step 4: Verify the Solution

The equation of the line is \(y = \frac{1}{2}x\). Confirm it is in slope-intercept form (\(y = mx + b\)) with slope \(\frac{1}{2}\) and \(y\)-intercept \(0\). It is indeed perpendicular to \(y = -2x + 6\), fulfilling the condition.

Key Concepts

Perpendicular Lines

Perpendicular lines are fascinating intersections in geometry. They meet at a right angle (90 degrees). This unique property makes perpendicular lines significant in mathematics, as they form the basis for defining right angles and constructing rectangles and squares. To determine whether two lines are perpendicular, we look at their slopes.

• If the product of their slopes is (-1), the lines are perpendicular.

In this exercise, we encountered a line with an equation: \(2x + y = 6\), and determined its slope to be \(-2\). The goal was to find a line passing through the origin and perpendicular to the given line. By understanding that perpendicular slopes are negative reciprocals, we could confidently proceed to the next steps.

Slope Calculation

The slope of a line is an essential concept in algebra and geometry. It signifies the steepness or the incline of the line. Mathematically, slope is often denoted by \(m\) and calculated as:

• \(m = \frac{{\text{change in } y}}{{\text{change in } x}}\)

To rewrite a line in slope-intercept form \(y = mx + b\), isolating \(y\) helps to identify this slope. In our exercise, for the equation \(2x + y = 6\), we isolated \(y\) to rewrite it as \(y = -2x + 6\). This revealed our slope \(m\) as \(-2\). Understanding how to manipulate equations to find slopes is a pivotal skill, as it easily reveals the incline of a line relative to the x-axis.

Negative Reciprocal

The concept of a negative reciprocal is fundamental when dealing with perpendicular lines. If two lines are perpendicular, their slopes must be negative reciprocals of each other. Let's break down what this means:

• The reciprocal of a number \(a\) is \(\frac{1}{a}\).
• The negative reciprocal is \(-\frac{1}{a}\).

For instance, if one line has a slope of \(-2\) like in our problem solving, the perpendicular line will have a slope of \(\frac{1}{2}\). Here, \(-2\) becomes \(-\frac{1}{(-2)} = \frac{1}{2}\). Utilizing the negative reciprocal effectively helps determine directions of perpendicular lines, and it's key in geometry for constructing right-angle intersections.