Problem 45 - Algebra 1

Question

Graph the equation. $$3 x+6 y=-18$$

Step-by-Step Solution

Answer

The line representing the equation \(3x + 6y = -18\) is a straight line which intersects the y-axis at -3 and has a slope of -0.5.

Step 1: Convert to Slope-Intercept Form

Re-arrange the given equation \(3x + 6y = -18\) to get it in the form \(y = mx + c\). Start by decreasing \(3x\) from both sides of the equation to isolate \(6y\). The result is \(6y = -3x - 18\). Divide each term of this equation by 6 to get \(y = -0.5x - 3\).

Step 2: Identify the Y-intercept and Slope

In the equation \(y = -0.5x - 3\), the y-intercept \(c\) is -3 and the slope \(m\) is -0.5. This signifies that the line crosses the y-axis at -3 and for each step in x, y decreases by 0.5.

Step 3: Plot the graph

Draw a straight line that intersects the y-axis at -3. Ensure that the line has a slope of -0.5. This implies that for every unit increase in x, y should decrease by 0.5 units. Make sure the line extends in both directions on the x-axis.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most useful ways to express a linear equation. It makes it easy to identify key components of the line and provides a straightforward way to graph it. This form is written as:
\[ y = mx + c \]
where:

• \( m \) is the slope of the line. The slope indicates how much the line rises or falls as it moves from left to right.
• \( c \) is the y-intercept of the line. This is the point where the line crosses the y-axis.

To convert an equation into slope-intercept form, you need to isolate \( y \) on one side of the equation. For example, given the equation \(3x + 6y = -18\), subtracting \(3x\) from both sides simplifies it to \(6y = -3x - 18\). Dividing every term by 6 then results in \(y = -0.5x - 3\). This transformation reveals the slope and y-intercept, making graphing a breeze.

Slope and Y-Intercept Identification

Once you have a linear equation in slope-intercept form \(y = mx + c\), identifying the slope and y-intercept becomes simple. These two components tell you a lot about the behavior and position of the line.
The slope \(m\) indicates the steepness and direction of the line:

• A positive slope means the line rises as it moves from left to right.
• A negative slope, as in our example \(-0.5\), means the line falls as it moves from left to right.

The y-intercept \(c\) is the point where the line crosses the y-axis. In this equation, the y-intercept is \(-3\). It tells you the location where x is zero. Understanding these components makes sketching and predicting the line's path easy.

Plotting Points on a Graph

Once you have identified the slope and y-intercept, plotting the linear equation on a graph becomes straightforward.
Here's how you can do it:

• Begin by marking the y-intercept on the graph. For the equation \(y = -0.5x - 3\), place a point at \(y = -3\) on the y-axis.
• Using the slope, plot the next point. Start at the y-intercept. Since the slope \(-0.5\) means for every 1 unit increase in x, y decreases by 0.5 units, move right by 1 unit on the x-axis and down by 0.5 units on the y-axis to place the next point.
• Draw a line through these points. Extend the line in both directions, as lines in mathematics continue indefinitely unless otherwise specified.

This method helps you clearly visualize how a linear equation is represented graphically.