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Chapter 11: Problem 173

In the following exercises, graph using the intercepts. $$ 3 x-2 y=6 $$

Short Answer

Expert verified
Graph the points (2, 0) and (0, -3) and draw a line through them.

Step by step solution

01

- Finding the x-intercept

To find the x-intercept, set y to 0 and solve for x in the equation. \[ 3x - 2(0) = 6 \] \[ 3x = 6 \] \[ x = 2 \] The x-intercept is (2, 0).
02

- Finding the y-intercept

To find the y-intercept, set x to 0 and solve for y in the equation. \[ 3(0) - 2y = 6 \] \[ -2y = 6 \] \[ y = -3 \] The y-intercept is (0, -3).
03

- Plotting the intercepts

Plot the intercepts on the coordinate plane. The points to plot are (2, 0) for the x-intercept and (0, -3) for the y-intercept.
04

- Drawing the line

Draw a straight line through the points (2, 0) and (0, -3) to graph the equation. Extend the line in both directions.

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Key Concepts

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

x-intercept
The x-intercept of a graph is where the line crosses the x-axis. This means the y-coordinate at this point is always zero.
To find the x-intercept for the equation given in the exercise, we set y to 0 and solve for x.

In our equation, we have:
  • The original equation: 3x - 2y = 6
  • Set y = 0: 3x - 2(0) = 6
  • Simplify to find x: 3x = 6, x = 2

This means the x-intercept is at the coordinate (2,0). This is the point where the line crosses the x-axis.
y-intercept
The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is always zero.
In the exercise, we find the y-intercept by setting x to 0 and solving for y.

For our equation:
  • The original equation: 3x - 2y = 6
  • Set x = 0: 3(0) - 2y = 6
  • Solve for y: -2y = 6, y = -3
This gives us the y-intercept at the coordinate (0, -3). This is the point where the line crosses the y-axis.
coordinate plane
The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at the origin (0,0).
In this exercise, once we have the x-intercept at (2,0) and the y-intercept at (0,-3), we plot these points on the coordinate plane.

Steps for plotting and drawing the line:
  • First, mark the x-intercept (2,0) on the x-axis.
  • Next, mark the y-intercept (0,-3) on the y-axis.
  • Draw a straight line passing through both points.
  • Extend this line in both directions to complete the graph.
By following these steps, you can visualize how the equation forms a straight line on the coordinate plane.

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