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Chapter 2: Problem 50

Find the \(x\) - and \(y\) -intercepts and use them to graph the following functions. $$-8 x+3 y=28$$

Short Answer

Expert verified
The x-intercept is \((-\frac{7}{2}, 0)\) and the y-intercept is \((0, \frac{28}{3})\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, we set the y-value to zero in the equation and solve for x. The given equation is \[-8x + 3y = 28\] Set \(y = 0\): \[-8x + 3(0) = 28\] This simplifies to \[-8x = 28\] Now, solve for x: \[x = \frac{28}{-8}\] \[x = -\frac{7}{2}\] Thus, the x-intercept is \((-\frac{7}{2}, 0)\).
02

Find the y-intercept

To find the y-intercept, we set the x-value to zero in the equation and solve for y. Again using the equation: \[-8x + 3y = 28\] Set \(x = 0\): \[-8(0) + 3y = 28\] This simplifies to: \[3y = 28\] Now, solve for y: \[y = \frac{28}{3}\] Thus, the y-intercept is \((0, \frac{28}{3})\).
03

Plot the intercepts and draw the graph

Using the intercepts calculated in the previous steps, plot the x-intercept \((-\frac{7}{2}, 0)\) and the y-intercept \((0, \frac{28}{3})\) on the coordinate plane. Then draw a straight line through these two points, as they are sufficient to determine the line of the equation.

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Key Concepts

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

x-intercept
The concept of the x-intercept is an essential part of understanding how to graph linear equations. The x-intercept refers to the point at which a line, plotted on a graph, crosses the x-axis.
At this point, the y-coordinate is always zero, as the line is not above or below the x-axis, but exactly on it.
To find the x-intercept for the equation \(-8x + 3y = 28\), we substitute \(y = 0\) and solve for \(x\).
  • This transforms the equation to \(-8x = 28\)
  • Then, solving for \(x\), we get \(x = -\frac{7}{2}\)
Hence, the x-intercept is \((-\frac{7}{2}, 0)\).Remember, at any x-intercept, the y-value is zero.
y-intercept
The y-intercept is equally important in graphing linear equations. It is the point where the line crosses the y-axis. At this intercept, the x-coordinate is always zero, because the line intersects the y-axis directly.
In our equation, \(-8x + 3y = 28\), we find the y-intercept by setting \(x = 0\) and solving for \(y\).
  • This changes the equation to \(3y = 28\)
  • Solving gives \(y = \frac{28}{3}\)
So, the y-intercept is \((0, \frac{28}{3})\).At any y-intercept, the x-value is zero. This point on the coordinate plane is crucial for plotting the line accurately.
coordinate plane
The coordinate plane is a flat, two-dimensional surface that helps us understand the positioning of points defined by a pair of numerical coordinates. These coordinates, expressed in the form \((x, y)\), identify the exact spot for any point on the plane.
When we graph a linear equation, such as \(-8x+3y=28\), the coordinate plane serves as our canvas.
  • The horizontal axis, or x-axis, runs left to right
  • The vertical axis, or y-axis, runs top to bottom
These two axes intersect at the origin, \((0, 0)\). Each point plotted, like the x- and y-intercepts, uses this coordinate system. Understanding the coordinate plane is fundamental.It helps organize and visualize the relationship between variables.
plotting points
Plotting points on the coordinate plane involves placing dots at specific locations based on their \((x, y)\) coordinates.
For the equation \(-8x + 3y = 28\), after calculating the \(x\) and \(y\) intercepts, plotting these points gives us a visual guide.
  • The x-intercept \((-\frac{7}{2}, 0)\) tells us where the line crosses the x-axis
  • The y-intercept \((0, \frac{28}{3})\) shows where it crosses the y-axis
With these two intercepts plotted, connect them with a straight line. This line is the graph of the equation.
Plotting is a key skill for visually analyzing equations and solutions.It allows us to translate mathematical calculations into clear visual representations.

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