Problem 60 - 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 perpendicular to the line \(y=-5 x\)

Step-by-Step Solution

Answer

The line's equation in standard form is \(x - 5y = 0\).

Step 1: Determine the slope of the given line

The equation of the given line is \(y = -5x\). The slope (m) of this line is \(-5\).

Step 2: Find the slope of the perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals. Therefore, if the original line's slope is \(-5\), the perpendicular line's slope \(m'\) is \(\frac{1}{5}\).

Step 3: Use the point-slope form of the line equation

The line must pass through the origin \((0,0)\) and have a slope of \(\frac{1}{5}\). The point-slope form is \[ y - y_1 = m'(x - x_1) \]Substitute \(x_1 = 0\), \(y_1 = 0\), and \(m' = \frac{1}{5}\):\[ y - 0 = \frac{1}{5}(x - 0) \]This simplifies to \(y = \frac{1}{5}x\).

Step 4: Convert the equation to standard form

The standard form of a line equation is \(Ax + By = C\). Rearrange \(y = \frac{1}{5}x\) to get:Multiply both sides by 5: \[ 5y = x \]Rearrange it to:\[ x - 5y = 0 \]So, the equation in standard form is \(x - 5y = 0\).

Key Concepts

Standard Form Equation

The standard form of a linear equation is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative. This form is particularly useful because it presents a neat, organized way to express a line without fractions or decimals. This form is also easy to grasp when solving systems of linear equations by elimination method or when determining intercepts.

As an example, let's consider the problem of converting the equation \(y = \frac{1}{5}x\) into standard form. To eliminate the fraction, multiply every term by 5, leading to \(5y = x\). Rearranging gives us \(x - 5y = 0\). Now, we have an equation where all coefficients are integers, fulfilling the standard form requirements.

To further grasp the concept:

• Ensure only integer coefficients in \(A\), \(B\), and \(C\).
• Place terms \(Ax\) and \(By\) on the left side.
• Maintain \(A\) as non-negative to follow convention.

Remember, this form makes it easier to find solutions to bigger problems involving systems of equations.

Slope

The slope of a line, often represented by the letter \(m\), measures the steepness or inclination of the line. It is defined as the rise over run, calculated as \(\frac{\text{change in } y}{\text{change in } x}\). In simpler terms, it tells you how much \(y\) increases for a unit increase in \(x\) along the line.

In our exercise, the initial line is \(y = -5x\) which has a slope of \(-5\). This negative slope indicates that the line goes downward as it moves to the right.

Unique to perpendicular lines is the concept of negative reciprocals. If a line has a slope of \(-5\), a line perpendicular to it will have a slope of \(\frac{1}{5}\). This means:

• Multiply an original slope by the perpendicular slope to get \(-1\).
• A positive slope portrays a line that goes upwards to the right.

Understanding slopes and their relationships is crucial for comprehensively solving geometric problems.

Point-Slope Form

Point-slope form is an efficient way of writing the equation of a line if you know its slope and one point on the line. The form is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope. \((x_1, y_1)\) is a point through which the line passes.

This form is particularly handy when the line does not cut the y-axis at an easy point or when working directly from a known point. Using the point-slope form to find an equation starts with understanding the line's slope and inserting the known point.

In the example, the line passes through the origin, \((0,0)\), with a slope of \(\frac{1}{5}\). Plug these values into the formula:
\[ y - 0 = \frac{1}{5}(x - 0) \]
The equation simplifies to \(y = \frac{1}{5}x\).

This point-slope form aids in deriving the linear equation, particularly when getting from raw geometric data to a usable equation form, enhancing quick calculations and transformations, eventually translating it to the standard form.