Problem 44 - Intermediate Algebra

Question

Contains the origin and is parallel to the line \(-2 x-9 y=4\)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{2}{9}x\).

Step 1: Understand the Given Line Equation

The given line equation is \(-2x - 9y = 4\). Our goal is to find a new line that is parallel to this line and passes through the origin (0,0).

Step 2: Convert the Equation to Slope-Intercept Form

Rearrange the line equation \(-2x - 9y = 4\) to the slope-intercept form \(y = mx + b\), where \(m\) is the slope.Start by subtracting \(-2x\) from both sides: \[-9y = 2x + 4\]Then, divide every term by \(-9\):\[y = -\frac{2}{9}x - \frac{4}{9}\]The slope \(m\) of this line is \(-\frac{2}{9}\).

Step 3: Identify the Slope of the Parallel Line

Since lines that are parallel share the same slope, the slope of the new line is also \(-\frac{2}{9}\).

Step 4: Use the Slope to Write the New Line Equation

We know the line is of the form \(y = mx + b\) and passes through the origin (0,0), which means when \(x = 0\), \(y = 0\). Substitute \(m = -\frac{2}{9}\) and the point (0,0) into the equation:\[y = -\frac{2}{9}x + b\]Since it passes through the origin, \(b = 0\), so the equation is:\[y = -\frac{2}{9}x\]

Step 5: Verify the Equation

To verify, check the line equation \(y = -\frac{2}{9}x\) passes through the origin. Substitute \(x = 0\) and check \(y\):\[y = -\frac{2}{9}(0) = 0\]This confirms the line passes through the origin and has the same slope as the original line \(-2x - 9y = 4\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular and easy-to-use way of expressing lines on a graph. It is written as \(y = mx + b\), where:

• \(m\) represents the slope of the line, which indicates how steep the line is.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

When rearranging an equation into this form, it's easier to read off the slope and the y-intercept directly. For instance, from the given equation \(-2x - 9y = 4\), when rearranged into \(y = -\frac{2}{9}x - \frac{4}{9}\), it makes it clear that the slope \(m\) is \(-\frac{2}{9}\), and the y-intercept \(b\) is \(-\frac{4}{9}\). This form is especially useful when graphing the line or when determining if two lines are parallel.

Linear Equation

A linear equation represents a straight line on a graph and can have different forms, such as slope-intercept form or standard form. The standard form is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. To understand the characteristics of the line, such as its slope and intercepts, the slope-intercept form is often more helpful.
In our exercise, we started with the linear equation \(-2x - 9y = 4\), and converted it to the slope-intercept form to find the slope. This transformation helps us understand the direction and steepness of the line, whether it increases or decreases as you move along the x-axis.

Origin

The origin in a coordinate plane is the point where both the x-axis and y-axis intersect, denoted as \((0,0)\). It is a pivotal reference point, especially when determining the y-intercept of a linear equation.

• Lines that pass through the origin have a y-intercept of 0.
• This makes their equation relatively simpler, since the y-intercept term, \(b\), is unnecessary: \(y = mx\).

In our solution, when identifying the equation of the new line that is parallel to \(-2x - 9y = 4\), knowing the line passes through the origin confirms that \(b = 0\). Therefore, the line is simply described by the equation \(y = -\frac{2}{9}x\), highlighting its direct relationship with the origin. This concept helps in quickly identifying and verifying equations of lines that cross through the origin point.