Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$8 x+y=0$$

Step-by-Step Solution

Answer

The slope of the line is -8 and the \(y\)-intercept is at (0, 0).

Step 1: Solve the equation for \(y\)

In the given equation \(8x + y = 0\), isolate \(y\) by subtracting \(8x\) both sides. This gives: \[y = -8x\]

Step 2: Identify the slope and \(y\)-intercept

From the slope-intercept form of the equation, we can easily identify the slope and the \(y\)-intercept. In this case, the equation \(y = -8x\) is of the same form where \(m\) (slope) is -8 and \(c\) (\(y\)-intercept) is 0. Therefore, the slope is -8 and the \(y\)-intercept is at (0, 0).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a special way of writing linear equations. It is written as \(y = mx + c\), where \(m\) is the slope of the line, and \(c\) is the \(y\)-intercept. This form is very useful because it allows us to quickly understand the behavior of a line just by looking at the equation. The slope \(m\) tells us how steep the line is, while the \(y\)-intercept \(c\) indicates where the line crosses the \(y\)-axis.

By converting any linear equation to slope-intercept form, we can easily visualize the line's position and steepness. For example, if the equation of the line is \(y = -8x\), it is already in slope-intercept form, with a slope \(m = -8\) and a \(y\)-intercept \(c = 0\). This makes it straightforward to graph the line or understand how it changes.

Solving for y

Solving for \(y\) means rearranging any given linear equation to express \(y\) solely in terms of \(x\). This is crucial when we want to convert an equation into slope-intercept form. Consider the equation \(8x + y = 0\). To solve for \(y\), we simply perform basic algebraic manipulations.

Here is a simple step-by-step guide to do so:

• Start by isolating the term containing \(y\). In this case, we can subtract \(8x\) from both sides to remove \(8x\) from the left-hand side. This results in \(y = -8x\).
• Now the equation is in the form \(y = mx + c\), which is the desired slope-intercept form.

This process of solving for \(y\) is fundamental as it allows us to directly identify the slope and intercept, making it easier to work with the equation.

Slope and y-Intercept

Understanding the slope and \(y\)-intercept is key to mastering linear equations.
The **slope** \(m\) of a line represents how much \(y\) changes for a unit change in \(x\). It tells us how steep the line is:

• A positive slope means the line is rising.
• A negative slope, like \(-8\) in \(y = -8x\), means the line is falling.
• A slope of 0 means the line is perfectly horizontal.

The **\(y\)-intercept** \(c\) shows where the line crosses the \(y\)-axis. It is the point \((0, c)\). In our example, the \(y\)-intercept is \(0\), meaning the line crosses the origin.

Identifying these two components from the slope-intercept form \(y = mx + c\) makes it easy to graph the line and predict its behavior. By knowing the slope and \(y\)-intercept, we can accurately describe the line's direction and starting point on a graph.