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Chapter 0: Problem 10

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") $$6 x+3 y=-9$$

Short Answer

Expert verified
Rewrite equation to y = -2x - 3, plot y-intercept (0, -3), use slope to plot (1, -5), draw the line.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

To graph the equation, first rewrite it in the form of y = mx + b. Start with the given equation: \[6x + 3y = -9\] Isolate y by first subtracting 6x from both sides: \[3y = -6x - 9\] Then, divide every term by 3 to solve for y: \[y = -2x - 3\]
02

Identify the Slope and Y-Intercept

From the equation \(y = -2x - 3\), identify the slope (m) and the y-intercept (b). The slope (m) is -2, and the y-intercept (b) is -3. This means the line crosses the y-axis at (0, -3) and has a slope of -2.
03

Plot the Y-Intercept

Begin by plotting the y-intercept on the graph. Find the point (0, -3) on the y-axis and place a dot there. This is the starting point for your line.
04

Use the Slope to Find Another Point

The slope (-2) tells us that for every 1 unit you move right (positive x direction), you move 2 units down (negative y direction). From the y-intercept (0, -3), move right 1 unit to (1, -3) and then move down 2 units to (1, -5). Plot the point (1, -5).
05

Draw the Line Through the Points

With the points (0, -3) and (1, -5) plotted, draw a straight line through these points. This is the graph of the equation \(6x + 3y = -9\). Extend the line in both directions and add arrows on both ends to indicate that the line continues infinitely.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
Understanding the slope-intercept form helps simplify graphing linear equations. The slope-intercept form is expressed as \(y = mx + b \), where:
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