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Chapter 2: Problem 68

Use intercepts to graph each equation. $$ 6 x-9 y-18=0 $$

Short Answer

Expert verified
The x-intercept is at (3, 0) and the y-intercept is at (0, -2). The line passing through these points represents the graph of the given equation.

Step by step solution

01

Find the x-intercept

To find the x-intercept, we must set y = 0 in the equation and then solve for x. By substituting y = 0 in 6x - 9y - 18 = 0, we get 6x - 18 = 0. Solving for x we get x = 3.
02

Find the y-intercept

To determine the y-intercept, we would set x = 0 in the equation and solve for y. By substituting x = 0 in 6x - 9y - 18 = 0, we get -9y - 18 = 0. Solving for y we get y = -2.
03

Plot the intercepts

On the graph, plot the x-intercept at (3, 0) and the y-intercept at (0, -2).
04

Draw the line

Connect the two points, which are the 'x' and 'y' intercepts, with a straight line.

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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 the point at which the line crosses the x-axis. It is important because it represents the value of 'x' when 'y' is zero. To find the x-intercept algebraically, you simply set the y-variable to zero in the equation and solve for 'x'. In the case of our equation, \( 6x - 9y - 18 = 0 \), we set \( y = 0 \) and solve for 'x', resulting in \( x = 3 \). This means our graph will cross the x-axis at the point \( (3, 0) \).

Understanding the x-intercept is crucial, as it can represent real-world scenarios like breaking even in a business model when cost equals revenue, or when a projectile lands back on the ground in physics.
Y-Intercept
Conversely, the y-intercept is where the line crosses the y-axis, denoting the value of 'y' while 'x' is zero. For calculating the y-intercept, set the x-variable to zero and solve for 'y'. Using our example, by making \( x = 0 \), the equation becomes \( -9y - 18 = 0 \), and solving yields \( y = -2 \). The point \( (0, -2) \) is where our line will intersect the y-axis.

The y-intercept often represents the starting point or initial condition in many applied problems, such as the initial amount of money before any transactions, or the starting height of a thrown ball.
Plotting Intercepts
To visualize a linear equation, plotting intercepts is a straightforward method. Start by marking the x-intercept, \( (3,0) \) in our case, on the graph. This point lies along the x-axis. Next, plot the y-intercept, \( (0, -2) \) on the y-axis. Both these points are crucial as they are where the line meets the axes, providing two reference points essential for drawing the line.

Tips for Accurate Plotting

Ensure that you plot these points accurately, use a ruler for precision, and double-check your coordinates to prevent any graphing error. The correct plotting of intercepts sets a solid foundation for representing the equation as a graph.
Linear Equation Graphing Steps
Graphing a linear equation involves systematic steps to represent the relationship between 'x' and 'y' visually. Here's a concise guide:
  1. Identify the equation to be graphed, like \( 6x - 9y - 18 = 0 \).
  2. Calculate the x-intercept by setting \( y = 0 \) and solving for 'x'.
  3. Find the y-intercept by setting \( x = 0 \) and resolving for 'y'.
  4. Plot both intercepts on a coordinate system, making sure the x-intercept is placed on the x-axis and the y-intercept on the y-axis.
  5. Draw a straight line through the plotted intercepts which represents the graph of the equation.

Remember, a ruler is crucial to get a straight line, symbolizing the consistent change between 'x' and 'y'. Graphing linear equations this way can help visualize how 'x' and 'y' interact with each other.

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