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Chapter 1: Problem 72

use intercepts to graph each equation. $$6 x-3 y+15=0$$

Short Answer

Expert verified
The x-intercept and y-intercept for the equation \(6x - 3y + 15 = 0\) are \(-2.5\) and \(5\) respectively.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve equation for \(x\). The equation becomes \(6x + 15 = 0\). Solving for \(x\) gives \(x = -\frac{15}{6} = -2.5\). So the x-intercept is \(-2.5\).
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve the equation for \(y\). The equation becomes \(-3y + 15 = 0\). Solving for \(y\) we obtain \(y = \frac{15}{3} = 5\). So, the y-intercept is \(5\).
03

Graph the Line using the intercepts

On a graph, plot the points \((-2.5, 0)\) and \((0, 5)\). Draw a line passing through these points. This line represents the given equation.

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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 is where a graph crosses the horizontal x-axis. This means the y-value is zero at this point, making it an essential step in graphing linear equations. To find the x-intercept, we need to set the y-value in the equation to zero and solve for x.
For example, with the equation \(6x - 3y + 15 = 0\), when we set \(y = 0\), the equation simplifies to \(6x + 15 = 0\).
Solving for \(x\) gives:
  • Subtract 15 from both sides: \(6x = -15\).
  • Divide by 6 to isolate \(x\): \(x = -\frac{15}{6} = -2.5\).
Thus, the x-intercept is at \((-2.5, 0)\). Knowing how to find this intercept helps plot one of the key points on the graph.
Unveiling the Y-Intercept
Moving to the y-intercept, we look at where the graph crosses the vertical y-axis. Here, the x-value is zero. Locating the y-intercept involves setting x to zero in the equation and solving for y.
With the equation \(6x - 3y + 15 = 0\), plug in \(x = 0\), giving us \(-3y + 15 = 0\).
By solving:
  • Subtract 15 from both sides: \(-3y = -15\).
  • Divide by -3 to solve for \(y\): \(y = 5\).
This tells us that the y-intercept is at \((0, 5)\). Identifying the y-intercept provides another crucial point for our graph.
Plotting Points to Graph
After finding both intercepts, you can use these points to draw the line that represents the linear equation.
The x-intercept at \((-2.5, 0)\) and the y-intercept at \((0, 5)\) provide two points. Plot these points on your graph.
Here's how to proceed:
  • Mark the x-intercept at \(-2.5\) along the x-axis.
  • Mark the y-intercept at \(5\) along the y-axis.
  • Use a ruler to draw a straight line through both points.
This forms the line of the equation \(6x - 3y + 15 = 0\). This technique of using intercepts for plotting is efficient and gives a clear visualization of the equation.

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Most popular questions from this chapter

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