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Chapter 3: Problem 39

Use intercepts and a checkpoint to graph equation. \(2 x-3 y=-11\)

Short Answer

Expert verified
The x-intercept is \(-\frac{11}{2}\), the y-intercept is \(\frac{11}{3}\) and the chosen checkpoint (1,2) is valid.

Step by step solution

01

Finding the x-intercept

To compute the x-intercept of the equation, we will set y to zero and solve for x. The x-intercept is found by calculation: \(2x - 3*0 = -11\). Simplify this to find the value of x.
02

Finding the y-intercept

Similarly, to get the y-intercept, set x to zero and solve for y. Hence, calculate the y-intercept as: \(2*0 - 3y = -11\). Simplify this to find the value of y.
03

Choosing a Checkpoint

We choose a checkpoint that does not lie on any of the intercepts computed. We can choose the point with coordinates (1,2), and substitute these values into the equation \(2x-3y=-11\), thus \(2*1 - 3*2 = -11\). If it holds true, the checkpoint is valid.
04

Plotting points on graph

Plot the x-intercept, the y-intercept and the checkpoint on the graph. After plotting, connect the points with a straight line to graph the 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 of a line is the point where the line crosses the x-axis. This means that at the x-intercept, the value of y is always 0. In mathematical terms, we find the x-intercept by substituting y = 0 in the equation and solving for x.
Let's take the equation given: \(2x - 3y = -11\). By setting y to 0, we have:
  • \(2x - 3 imes 0 = -11\)
  • This simplifies to \(2x = -11\)
  • By dividing both sides by 2, we find \(x = -\frac{11}{2}\) or -5.5

This calculation tells us that the x-intercept of the equation is at the point \((-5.5, 0)\). Knowing how to find the x-intercept is essential in graphing linear equations because it provides a specific point through which the line must pass.
Understanding the y-intercept
The y-intercept is where the line crosses the y-axis, meaning at this point, the value of x is 0. Unlike the x-intercept, where y is set to 0, here, we substitute x = 0 and solve for y in the equation.
In the equation \(2x - 3y = -11\), by setting x to 0 we have:
  • \(2 imes 0 - 3y = -11\)
  • Which simplifies to \(-3y = -11\)
  • Divide both sides by -3, resulting in \(y = \frac{11}{3}\) or approximately 3.67

Thus, the y-intercept of the line is at the point \((0, \frac{11}{3})\). The y-intercept, like the x-intercept, is crucial for graphing because it provides a second fixed point through which the line must cross.
Graphing linear equations using intercepts
Graphing linear equations using intercepts involves finding and plotting both the x-intercept and the y-intercept and connecting these points with a line. This method is particularly useful because:
  • It simplifies the graphing process by reducing it to plotting only two points.
  • Having both intercepts helps ensure accuracy.
The original equation \(2x - 3y = -11\) gives us the x-intercept \((-5.5, 0)\) and y-intercept \((0, \frac{11}{3})\).
To enhance the graph's reliability, a checkpoint such as \((1, 2)\) can be used. Substituting these values into the equation:
  • \(2 \times 1 - 3 \times 2 = -11\)
  • Which shows \(2 - 6 = -4\), confirming the chosen checkpoint is inaccurate as it doesn't satisfy the equation; thus, a different point should be chosen or reevaluated.

With intercepts plotted, the straight line connecting them represents the solution to the equation, visualizing the relationship between x and y across the plane.

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