Problem 62 - Introductory Algebra for College Students

Question

Write an equation in the form \(y=m x+b\) of the line that is described. The line has the same \(y\) -intercept as the line whose equation is \(2 y=6 x+8\) and is parallel to the line whose equation is \(4 x+4 y=20\).

Step-by-Step Solution

Answer

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

Step 1: Calculate the y-intercept

First, rearrange the equation \(2y = 6x + 8\) to the form \(y = mx + b\). This is achieved by dividing every term in the equation by 2, yielding \(y = 3x + 4\). The y-intercept \(b\) is the constant in the equation, thus \(b = 4\).

Step 2: Calculate the slope

Since the line is parallel to \(4x + 4y = 20\), it shares the same slope. Rearrange \(4x + 4y = 20\) by isolating \(y\). Subtract \(4x\) from each side and then divide each term by 4: \(y = -x + 5\). The slope \(m\) is the coefficient of \(x\), hence \(m = -1\).

Step 3: Formulate the Equation

We now have both necessary elements for the equation in slope-intercept form, \(y = mx + b\). By substituting \(m = -1\) and \(b = 4\) into the equation, the equation of the line is found to be \(y = -x + 4\).

Key Concepts

y-intercept

The concept of the y-intercept is a cornerstone of understanding linear equations. In any line equation in the form of \(y = mx + b\), the y-intercept is represented by \(b\). This is the point where the line crosses the y-axis, which is significant because it shows us the value of \(y\) when \(x = 0\). The y-intercept gives us a starting point for graphing a line.
To find the y-intercept from an equation like \(2y = 6x + 8\), we rearrange it into \(y = mx + b\) form. Divide everything by 2 to clear the coefficient of \(y\), resulting in \(y = 3x + 4\). Here, the constant term \(4\) is the y-intercept, signifying that the line will cross the y-axis at \(y = 4\). This information helps in sketching the line on a graph without needing additional points.

slope

The slope of a line indicates its steepness and direction. In the equation \(y = mx + b\), the slope is \(m\). It tells us how much \(y\) changes for every change in \(x\). If \(m\) is positive, the line rises as it moves from left to right. If \(m\) is negative, like in our example with \(m = -1\), the line falls. This means for every one unit increase in \(x\), \(y\) decreases by one unit.
Slope is crucial for identifying parallel lines. Lines that are parallel share the same slope. For instance, the given line \(4x + 4y = 20\) can be rearranged to \(y = -x + 5\), revealing a slope \(m = -1\). Therefore, any line parallel to this must also have a slope of \(-1\). This ensures that the lines move in the same direction, maintaining a consistent separation or distance between them.

parallel lines

Parallel lines run alongside each other, never meeting, akin to railway tracks. This happens because they have the same slope. In algebra terms, to check if two lines are parallel, you look at the slope \(m\) in their equations \(y = mx + b\). If two lines have identical \(m\) values, they are parallel.
Given an equation like \(4x + 4y = 20\), rearranging it to \(y = -x + 5\) reveals a slope of \(-1\). A line parallel to this has the same slope. With our task, because the new line must be parallel, we also set its slope to \(-1\). The result is a line that runs parallel to \(y = -x + 5\), keeping the structure consistent with any parallel strategies or constructions in geometry or architecture. Parallel lines are foundational in creating designs that need consistency and symmetry.