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Chapter 1: Problem 70

use intercepts to graph each equation. $$3 x+5 y+15=0$$

Short Answer

Expert verified
The x-intercept is found when y=0 in the equation and calculated as \(x=-5\). The y-intercept is found when x=0 and calculated as \(y=-3\). By plotting these points in the graph and drawing a line through these points, the graph of the equation \(3x + 5y + 15 = 0\) can be obtained.

Step by step solution

01

Find the x-intercept

The x-intercept is the value of x when y is zero. To find the x-intercept, set y = 0 in the equation \(3x + 5(0) + 15 = 0\), therefore, solve for x to get the x-intercept.
02

Find the y-intercept

The y-intercept is the value of y when x is zero. To find the y-intercept, set x = 0 in the equation \(3(0) + 5y + 15 = 0\), therefore, solve for y to get the y-intercept.
03

Plot the intercepts on the graph

Once you have the x and y intercepts, plot these two points on the graph. The x-intercept should be plotted on the x-axis and the y-intercept should be plotted on the y-axis.
04

Draw the line

Finally, draw a line through the two points plotted on the graph. This is the line that represents the equation \(3x + 5y + 15 = 0\) on the graph.

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Key Concepts

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

Understanding the x-intercept
The x-intercept of a linear equation is a crucial point where the graph of the equation crosses the x-axis. At the x-intercept, the y-value is always zero because it lies exactly on the x-axis. To find the x-intercept in the given equation \(3x + 5y + 15 = 0\), follow these steps:
  • First, replace \(y\) with 0 in the equation, since at the x-intercept, y = 0.
  • This simplifies the equation to \(3x + 15 = 0\).
  • To solve for \(x\), subtract 15 from each side, resulting in \(3x = -15\).
  • Finally, divide both sides by 3, yielding \(x = -5\).
Thus, the x-intercept is the point \((-5, 0)\). Remember, an x-intercept is where the line meets the x-axis, making it a critical point for graphing linear equations.
Discovering the y-intercept
The y-intercept represents the point where a line crosses the y-axis. At this specific point, the x-value is always zero. Finding the y-intercept of the equation \(3x + 5y + 15 = 0\) involves these steps:
  • Set \(x\) to zero because the y-intercept occurs when x = 0.
  • This transforms the equation into \(5y + 15 = 0\).
  • Next, solve for \(y\) by subtracting 15 from both sides, which gives \(5y = -15\).
  • Divide both sides by 5, resulting in \(y = -3\).
So, the y-intercept is the point \((0, -3)\). Understanding the y-intercept is fundamental in graphing as it provides a starting point for drawing the line on a graph.
Plotting points and graphing the line
Plotting points, such as the intercepts, is the last step toward visualizing your linear equation. With the x-intercept \((-5, 0)\) and the y-intercept \((0, -3)\), you have all you need to draw the graph of the equation \(3x + 5y + 15 = 0\). Here鈥檚 how to do it:
  • Begin by placing the x-intercept on the x-axis at point \((-5, 0)\).
  • Next, mark the y-intercept on the y-axis at point \((0, -3)\).
Once the points are plotted, draw a line that passes through both intercepts. This line extends infinitely in both directions and represents all solutions to the equation.
Graphing linear equations by plotting the intercepts is an effective method because it is simple and straightforward. It helps you see how the equation behaves on a graph and comprehend its slope and visual representation.

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