Problem 34 - Thinking Mathematically

Question

In Exercises 33-40, 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 line. \(2 x+y=0\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y=-2x\). The slope is -2 and the y-intercept is 0. The graph would start from the point (0,0) and slope downwards to the right.

Step 1: Converting to slope-intercept form

The given equation is \(2x + y = 0\). In order to get y by itself, you'll need to subtract \(2x\) from both sides. This gives you the equation in slope-intercept form: \(y = -2x + 0\).

Step 2: Identifying the slope and y-intercept

From the slope-intercept form, it can be clearly seen that the slope (m) is -2 and the y-intercept (b) is 0. So, the line intersects the y-axis at 0 and for every step of 1 in the x direction, it goes 2 steps down (negative slope) in the y direction.

Step 3: Graphing the line

To graph this line, you can start by putting a point on 0 in the y-axis (which is the y-intercept). This line slope is negative, which means it is decreasing. For each unit you move to the right on the x-axis, move two units down on the y-axis to reflect the slope of -2. Draw a line through these points to represent the line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is like having a roadmap for graphing lines. This specific form is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) indicates the y-intercept. The power of this format lies in its simplicity; it allows you to quickly identify two critical attributes of a line without rearranging the equation.

For instance, let's look at a basic equation: \(2x + y = 0\). To transform this into the slope-intercept form, we need to isolate the variable \(y\). Here's how it's done:

• Subtract \(2x\) from both sides of the equation to get \(y = -2x\).
• There's no constant term added to \(-2x\), which implies that the y-intercept, \(b\), is \(0\), resulting in the final form of \(y = -2x + 0\).

By converting to this form, it makes it significantly easier to graph the equation since the slope and y-intercept are immediately visible.

Slope and Y-intercept

The slope and y-intercept are the dynamic duo of linear equations, providing all the information needed to graph a line swiftly. The slope, often referred to by the letter \(m\), indicates the line’s steepness and direction. A positive slope ascends from left to right, while a negative slope descends.

Interpreting Slope

If a line has a slope of \(-2\), as in our example \(y = -2x + 0\), it shows that for every single unit increase in \(x\), the value of \(y\) decreases by 2 units. It’s a bit like walking down a hill; each step forward is also a step downward.

Identifying the Y-intercept

As for the y-intercept, represented by \(b\) in our equation, it's the point where the line crosses the y-axis. With a y-intercept of \(0\), our line kisses the origin, the point \((0,0)\), highlighting where the graphing starts. Knowing these two components, one can plot the equation on the graph with ease and precision.

Graphing Lines

The art of graphing lines on a coordinate plane may seem daunting at first, but with a grasp of the slope and y-intercept, it becomes a straightforward task. Graphing involves taking the y-intercept and slope we identified and turning them into a visual representation of the equation.

Starting with the Y-intercept

For the equation \(y = -2x\), we start at the origin, since the y-intercept, \(b\), is \(0\). You'll plot this initial point on the y-axis.

Applying the Slope

Then, using the slope of \(-2\), plot additional points. Since it's negative, move to the right 1 unit (a standard move in the positive x-direction), then go down 2 units (reflected by our negative slope). Repeat this process to ensure accuracy and connect the dots to draw the line.

This method allows for a quick, efficient way to produce the graphical representation of any linear equation you encounter. By mastering these steps, you can transition from the abstract concepts of slope and y-intercept to the concrete creation of a line on a graph.