Problem 39 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Put the equation in slope-intercept form by solving for \(y .\) b. Identify the slope and the \(y\) -intercept. c. Use the slope and y-intercept to graph the equation. $$3 x+y=0$$

Step-by-Step Solution

Answer

The slope-intercept form of given equation is \(y = -3x\). The slope is -3 and the y-intercept is 0.

Step 1: Put the equation in slope-intercept form

Start by isolating \(y\) in the equation \(3x + y = 0\). This can be achieved by subtracting \(3x\) from both sides of the equation. This gives: \(y = -3x\).

Step 2: Identify the slope and the y-intercept

With the equation in the form \(y = -3x\), it can be observed that the number multiply with \(x\) (which is -3) is the slope of the line and the constant term (which is not present here, implying it to be 0) is the y-intercept.

Step 3: Use the slope and y-intercept to graph the equation

Start by plotting the y-intercept on the y-axis. Since the y-intercept is 0, mark a point at the origin. Now, the slope of the line is -3, which means for every increase of 1 unit in \(x\), \(y\) decreases by 3 units. So go right 1 unit from the origin and down 3 units and plot a point. Draw a line passing through the two points which will represent the equation.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that allows us to visualize equations as lines on a coordinate plane. Every linear equation can be written in the form of \(y = mx + b\), known as the slope-intercept form, where the line is defined by two key components: the slope \(m\) and the y-intercept \(b\). To graph a linear equation like \(y = -3x\), we'll first highlight where the line crosses the y-axis, which is facilitated by the y-intercept. Then we use the slope to find another point on the line.
Start by plotting the y-intercept on the graph, which is a simple and clear way to get started. You can then use the slope to identify additional points that the line passes through, ensuring accuracy in your graph. This approach makes graphing both visual and practical, offering a clear representation of the relationship between the variables in your equation.

• First, identify the equation’s slope and intercept.
• Next, plot the y-intercept on the y-axis.
• Finally, use the slope to find another point and draw the line.

Approaching graphing in this consistent way will enhance your ability to interpret and understand linear relationships effectively.

Slope

The slope of a line tells us how steep the line is. It is often referred to as "rise over run." This simply means it describes how much the line moves up or down (rise) for a one unit move to the right (run). In the slope-intercept equation \(y = mx + b\), the slope is represented by the coefficient \(m\).
In the equation \(y = -3x\), the slope is \(-3\). This means that for each one unit you move to the right along the x-axis, you should move down three units along the y-axis. A negative slope like \(-3\) indicates that the line tilts downwards from left to right. It's crucial to understand slope as it defines the direction and steepness of the line.

• A positive slope means the line rises as you move from left to right.
• A negative slope means the line falls as you move from left to right.
• A zero slope results in a horizontal line, indicating no change.

Being comfortable with slope can greatly improve your ability to quickly analyze and graph linear equations.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is the constant \(b\). This value signifies the point where the value of \(x\) is zero, helping us identify the starting point of our line on the graph.
For the equation \(y = -3x\), you will notice that there is no constant term, which implicitly means the y-intercept is 0. This tells us our line crosses the y-axis at the origin, or the point \((0, 0)\).

• The y-intercept provides a starting point for graphing.
• If \(b\) is positive, the line crosses above the origin.
• If \(b\) is negative, the line crosses below the origin.
• If \(b\) is zero, the line goes through the origin itself.

Understanding the y-intercept not only facilitates easy graphing but also provides insight into the equation's initial conditions.