Problem 58 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The equation of the line is \(y = -2x - 4\)

Step 1: Calculate the slope of the given line

Convert the given line's equation, \(2x+y=8\), to the slope-intercept form to identify the slope. To do so, solve for \(y\) by subtracting \(2x\) from both sides of the equation. This results in \(y=-2x+8\). The coefficient of \(x\) here, \(-2\), is the slope of the given line.

Step 2: Use the slope for our line

Since our line is parallel to the given line, it will have the same slope, which we've found to be \(-2\). Therefore, \(m=-2\) for the equation of our line.

Step 3: Use the y-intercept for our line

The exercise provides the y-intercept \(-4\) for our line. Therefore, \(b=-4\) in the equation of our line.

Step 4: Write the equation of our line

Substitute the values of \(m\) and \(b\) we found into the equation \(y=mx+b\) to get the equation of our line, which is \(y=-2x-4\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental for writing linear equations efficiently. It's expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. The slope corresponds to the steepness and direction of the line, positive slopes go upwards, while negative slopes point downwards. To convert any linear equation to the slope-intercept form, you'll need to isolate \( y \) on one side of the equation.

For instance, given the equation \( 2x + y = 8 \), through simple manipulations by subtracting \( 2x \) from both sides, we get \( y = -2x + 8 \). Here, the coefficient of \( x \), which is \( -2 \), is the slope, and the constant term, \( 8 \), is the y-intercept.

Parallel Lines

Parallel lines are straight lines in the same plane that never meet, no matter how long they are extended. A key feature of parallel lines is that they have identical slopes. When given a line like \( y = -2x + 8 \) and asked to find a line parallel to it, the new line will have the same slope, in this case, \( -2 \). Knowing a line's equation allows you to quickly identify the slope and write the equation of another line parallel to it, provided you have a point or the y-intercept.

Y-intercept

The y-intercept is where a line crosses the y-axis on a graph. It's where the value of \( x \) is zero. This point is represented by the \( b \) in the slope-intercept equation, \( y = mx + b \). If you're given the y-intercept outright, as in the exercise where the y-intercept is \( -4 \), it significantly simplifies writing your linear equation, since you already have the value of \( b \) for the equation \( y = mx + b \).

Linear Equations

Linear equations describe straight lines and can be written in various forms, including slope-intercept form. A linear equation in two variables like \( x \) and \( y \) represents a line on the Cartesian plane. When you're given specific characteristics of a line, such as its slope and y-intercept, you can construct the linear equation by plugging these variables into the standard form, which is most commonly the slope-intercept form for ease and clarity.

For the problem at hand, having identified the slope \( m = -2 \) and y-intercept \( b = -4 \), you combine these to write the final linear equation: \( y = -2x - 4 \). This is a straightforward process requiring algebraic manipulation and a grasp of the concepts of slope and y-intercept, which are the building blocks of linear equations.