Problem 55 - 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 \(4 x-7 y=3\)

Step-by-Step Solution

Answer

The equation of the line is \(4x - 7y = 0\).

Step 1: Identify Slope of the Given Line

The given line is in standard form: \(4x - 7y = 3\). To find the slope, we first convert it into slope-intercept form \(y = mx + b\), where \(m\) represents the slope. Rearranging, we have:\[-7y = -4x + 3\]\[y = \frac{4}{7}x - \frac{3}{7}\]The slope \(m\) of the original line is \(\frac{4}{7}\).

Step 2: Use the Same Slope for the Parallel Line

Lines that are parallel have the same slope. Since we need a line parallel to \(4x - 7y = 3\), the parallel line will also have a slope of \(\frac{4}{7}\).

Step 3: Determine Equation Using Given Point

Use the point-slope form of the equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point (0,0), which is the origin, and \(m = \frac{4}{7}\). Substitute these values:\[y - 0 = \frac{4}{7}(x - 0)\]\[y = \frac{4}{7}x\]

Step 4: Convert to Standard Form

The standard form of a linear equation is \(Ax + By = C\). We need to eliminate fractions:\[7y = 4x\]Rearranging gives the standard form:\[4x - 7y = 0\].

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and the direction it moves on the coordinate plane. It is defined as the ratio of the vertical change, known as the "rise," to the horizontal change, the "run," between two points on the line. The formula for calculating slope, often denoted as \( m \), is:

• \( m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \)

In the context of linear equations, when expressed in slope-intercept form \( y = mx + b \), \( m \) directly gives us the slope of the line.
Calculating the slope is a crucial first step in many line-related problems, as it determines how the line behaves as it moves across the graph. Understanding slope helps in identifying whether lines are parallel, perpendicular, or intersecting.

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts. For two lines to be parallel, their slopes must be identical.
For example, given lines with equations:

• \( y = \frac{4}{7}x + b_1 \) and \( y = \frac{4}{7}x + b_2 \)

Since both lines have the slope \( \frac{4}{7} \), they are parallel.
Identifying lines as parallel is essential for solving geometric problems and ensuring accuracy when drawing graphs. This concept is also key when using algebra to solve real-world problems involving parallel paths or patterns.

Standard Form of a Line

The standard form of a line is a way of writing linear equations in the format \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative integer.
This form is particularly useful for certain algebraic operations, like finding intersections or comparing equations easily. To convert an equation into standard form, you often need to eliminate fractions and rearrange the terms. Let's see an example:

• Transform \( y = \frac{4}{7}x \) into standard form.

Start by removing fractions by multiplying the entire equation by 7: \( 7y = 4x \).
Then, rearrange to get \( 4x - 7y = 0 \).
Understanding the standard form is helpful for solving many types of algebra problems and gives a structured approach to manipulate and compare equations efficiently.