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Chapter 2: Problem 71

Use intercepts to graph the each equation. $$8 x-2 y+12-0$$

Short Answer

Expert verified
The x-intercept is \((-1.5, 0)\) and y-intercept is \((0,6)\). By connecting these points, the equation \(8x - 2y + 12 = 0\) is represented as a straight line on the graph.

Step by step solution

01

Finding the x-intercept

To find the x-intercept, set the parameter \(y\) equal to 0 in the equation and solve for \(x\). The equation becomes \(8x + 12 = 0\). The solution is \(x = -\frac{3}{2}\).
02

Finding the y-intercept

To find the y-intercept, set the parameter \(x\) equal to 0 in the equation and solve for \(y\). The equation becomes \(-2y + 12 = 0\). The solution is \(y = 6\).
03

Plotting the Graph

Start by plotting the x-intercept at \((-1.5,0)\) and then plot the y-intercept at \((0, 6)\). Connect these points with a straight line to represent the equation \(8x - 2y + 12 = 0\) on the graph.

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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 is a key concept when dealing with linear equations and their graphs. It's the point where the graph of the equation crosses the x-axis. At this point, the value of y is always 0. For the given equation \( 8x - 2y + 12 = 0 \), you find the x-intercept by setting \( y = 0 \).
  • Substitute \( y = 0 \) into the equation: \( 8x + 12 = 0 \).
  • Solve for \( x \): \( 8x = -12 \) gives \( x = -\frac{3}{2} \).
This means the x-intercept is at the point \( (-1.5, 0) \). Understanding this concept is crucial because it gives you a specific point to start graphing a linear equation. It's one of the two critical points needed to graph a line using intercepts.
Y-Intercept
The y-intercept of a linear equation is where the graph crosses the y-axis. Here, the value of \( x \) is always 0, which simplifies finding this intercept.
  • To find the y-intercept of \( 8x - 2y + 12 = 0 \), set \( x = 0 \).
  • The equation simplifies to \( -2y + 12 = 0 \).
  • Solve for \( y \): \( -2y = -12 \) gives \( y = 6 \).
Thus, the y-intercept is \( (0, 6) \). This point provides another anchor for drawing the graph. Together with the x-intercept, you can draw the line representing the equation, greatly simplifying the process of graphing.
Coordinate Plane
A coordinate plane is a two-dimensional surface formed by two perpendicular lines known as axes. The horizontal axis is called the x-axis, while the vertical axis is the y-axis. These axes intersect at the origin, denoted as \( (0,0) \).
  • Each point on the plane can be specified by a pair of numbers, \( (x,y) \), known as coordinates.
  • Coordinates indicate the position on the plane relative to the origin.
When you plot the x-intercept \( (-1.5, 0) \) and the y-intercept \( (0, 6) \) on the coordinate plane, you can connect these points to graph the linear equation. The coordinate plane is essential as it provides a visual representation of equations and solutions, allowing us to clearly see the relationships and intersections of lines.
Linear Equation Solutions
Linear equations, like \( 8x - 2y + 12 = 0 \), have solutions that can be plotted on a graph as a straight line. The solutions are the set of all points \( (x, y) \) that satisfy the equation.
  • Each solution corresponds to a point on the line.
  • The process of solving involves finding meaningful points such as intercepts, which can be graphed to identify the line.
In this exercise, the x-intercept and y-intercept are both solutions that help us establish the line visually. By connecting these intercepts, you draw the line that represents all potential solutions for the equation. Understanding how to find and use these solutions is crucial for graphing and interpreting linear equations accurately.

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Most popular questions from this chapter

In Exercises \(105-108,\) you will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used, \(x .\) Then find and interpret \(S(30)\)

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+3)^{2}+(y-2)^{2}=4 $$

Suppose that a function \(f\) whose graph contains no breaks or gaps on \((a, c)\) is increasing on \((a, b),\) decreasing on \((b, c)\) and defined at \(b\). Describe what occurs at \(x=b\). What does the function value \(f(b)\) represent?

a. Graph the functions \(f(x)=x^{n}\) for \(n=2,4,\) and 6 in a \([-2,2,1]\) by \([-1,3,1]\) viewing rectangle. b. Graph the functions \(f(x)=x^{n}\) for \(n=1,3,\) and 5 in a \([-2,2,1]\) by \([-2,2,1]\) viewing rectangle. c. If \(n\) is positive and even, where is the graph of \(f(x)=x^{n}\) increasing and where is it decreasing? d. If \(n\) is positive and odd, what can you conclude about the graph of \(f(x)=x^{n}\) in terms of increasing or decreasing behavior? e. Graph all six functions in a \([-1,3,1]\) by \([-1,3,1]\) viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?

The function $$ f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95 $$ models the number of annual physician visits, \(f(x),\) by a person of age \(x .\) Graph the function in a \([0,100,5]\) by \([0,40,2]\) viewing rectangle. What does the shape of the graph indicate about the relationship between one's age and the number of annual physician visits? Use the \([\mathrm{TABLE}]\) or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
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