Problem 40 - 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. $$2 x+y=0$$

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = -2x\). The slope is \(-2\) and the y-intercept is \(0\).

Step 1: Rearrange the equation to slope-intercept form

Start by rearranging the equation \(2x + y = 0\) into slope-intercept form. This is done by isolating \(y\) on one side. So subtract \(2x\) from both sides: \(y = -2x\).

Step 2: Identify the slope and y-intercept

Now, this equation identifies the slope \(m\) and y-intercept \(b\). In this equation, \(m=-2\) (which is the coefficient of \(x\) ), and \(b=0\) (which is the constant term).

Step 3: Graph the equation

You will start graphing by marking the y-intercept, which is \(0\) in this case. Because the slope is \(-2\), from the y-intercept, you will go down \(2\) units (negative slope) and over to the right by \(1\) unit, for every subsequent point. Repeat this process to get enough points and then join all the points to make a straight line on a graph.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for anyone beginning to explore the world of linear equations.

The slope-intercept form of a linear equation is expressed as: \( y = mx + b \), where \(m\) represents the slope, and \(b\) represents the y-intercept. The slope is a measure of how steep the line is, and the y-intercept is the point where the line crosses the y-axis.

To put an equation into slope-intercept form, you need to solve for \(y\) and make it the subject of the equation. Like in the provided exercise, \(2x + y = 0\) is transformed into \(y = -2x + 0\), which reveals that the slope \(m\) is \( -2\) and the y-intercept \(b\) is \(0\). The slope being negative indicates that the line slopes downward as you move from left to right across the graph.

Identifying Slope and Y-Intercept

The first step towards plotting a linear equation is to identify its slope and y-intercept from the slope-intercept form. The slope, \(m\), indicates the rate at which the line rises or falls as you move along the x-axis.

The y-intercept, \(b\), is the value of \(y\) when \(x=0\). It's the point where the line hits the y-axis, which can be plotted right away on a graph. Once you know these two vital pieces of information, you can begin to sketch the behavior of the line.

In our exercise, \(y = -2x\) makes it clear that our slope, \(m\), is \( -2\), and our y-intercept, \(b\), is \(0\). This means that for each single-unit increase in \(x\), \(y\) decreases by 2 units. Also, the line touches the y-axis at the origin (0,0).

Plotting Linear Equations

Plotting linear equations involves drawing a straight line that represents all the solutions to the equation.

Start by locating the y-intercept on the graph, which is your initial point. From there, use the slope to determine the direction and steepness of the line. Remember, the slope is a ratio that indicates how many units to move up or down for every unit you move to the right along the x-axis.

In the provided exercise, you would begin at the origin (0,0), because the y-intercept is zero. Since the slope is \( -2 \), you move down 2 units vertically for every 1 unit you move to the right horizontally. Connect these points to form a straight line, and this line is the graphical representation of the equation \(y = -2x\).