Problem 56 - Elementary Algebra

Question

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the origin and is parallel to the line \(-2 x-9 y=4\)

Step-by-Step Solution

Answer

The standard form of the line is \(2x + 9y = 0\).

Step 1: Identify the Slope of the Given Line

First, convert the equation \(-2x - 9y = 4\) into slope-intercept form (\(y = mx + b\)) by solving for \(y\). Start by adding \(2x\) to both sides:\[-9y = 2x + 4\]Now divide each term by \(-9\):\[y = -\frac{2}{9}x - \frac{4}{9}\]The slope (\(m\)) of the line is \(-\frac{2}{9}\). A line parallel to this will have the same slope.

Step 2: Use the Slope-Point Form

Since the line must contain the origin, use the point-slope form of the equation \(y - y_1 = m(x - x_1)\) where \((x_1, y_1) = (0, 0)\) and \(m = -\frac{2}{9}\).Plugging in the values:\[y - 0 = -\frac{2}{9}(x - 0)\]This simplifies to:\[y = -\frac{2}{9}x\].

Step 3: Convert to Standard Form

To convert \(y = -\frac{2}{9}x\) into standard form, rearrange terms so that all terms are on one side of the equation, eliminating fractions:First, multiply each term by 9 to eliminate the fraction:\[9y = -2x\]Rearrange the terms:\[2x + 9y = 0\]This is the standard form of the equation.

Key Concepts

Slope-Intercept Form

To understand the connection between lines and their equations, it's key to grasp the **slope-intercept form**. This form is represented as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept—the point where the line crosses the y-axis.
The slope \(m\) indicates how steeply the line rises or falls as it moves along the x-axis. If \(m\) is positive, the line inclines upwards. If \(m\) is negative, the line declines downwards. The slope is calculated as "rise over run," or the change in \(y\) divided by the change in \(x\) between two points on the line.
Here’s a quick breakdown of the steps to convert from standard form \(-2x - 9y = 4\) to slope-intercept form:

• Resolve for \(y\) by isolating it on one side of the equation.
• Move any \(x\) term to the opposite side by performing the inverse operation.
• Divide all terms by the coefficient of \(y\) to solve for \(y\).

Understanding this form allows you to quickly determine the behavior of a line just by looking at an equation.

Parallel Lines

Lines that never meet, no matter how far they extend, are called **parallel lines**. In the context of linear equations, parallel lines have identical slopes. This means they rise and run in the same proportion.
When you have an equation, such as \(-2x - 9y = 4\), and need a line parallel to it, the slope of the new line must be the same. We found the slope to be \(-\frac{2}{9}\) when this equation was converted to slope-intercept form. A parallel line retains this slope regardless of where it crosses the y-axis.
To create the equation for a new line that is parallel:

• Keep the slope \(m\) the same as the original line.
• Choose a different point for the line to pass through, which could be given or your own choice.

This principle of matching slopes maintains the lines' parallel relationship.

Point-Slope Form

A particularly useful form for writing line equations is the **point-slope form**. This is expressed as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a specific point on the line, and \(m\) is the line's slope.
In our exercise, we needed a line through the origin \((0, 0)\) that is parallel to a given line. With a known point and slope, the point-slope form easily constructs the desired equation:

• Insert the known point's coordinates \((x_1, y_1)\) into the equation.
• Use the parallel line's slope \(m = -\frac{2}{9}\).

So, plugging the origin into our formula, we have:\[y - 0 = -\frac{2}{9}(x - 0)\]
This simplifies to \(y = -\frac{2}{9}x\), perfectly aligning the parallel point-slope form with our needs. It's a straightforward technique that proves especially handy when dealing with specific points and slopes.