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Chapter 10: Problem 6

In Exercises 1 to \(8,\) draw the graph of each equation. Name any intencepts. $$3 y-9=0$$

Short Answer

Expert verified
The graph is a horizontal line at \(y=3\) with a y-intercept at \((0,3)\).

Step by step solution

01

Understand the Equation

The given equation is \(3y - 9 = 0\). This is a linear equation in the variable \(y\). Our goal is to express it in the form \(y = mx + b\), which represents a horizontal line in this case.
02

Solve for y

To express the equation in terms of \(y\), we can solve for \(y\). Add \(9\) to both sides:\[3y = 9\]Then, divide both sides by \(3\):\[y = 3\].This tells us that \(y\) is always \(3\), which is the equation of a horizontal line.
03

Identify and Name Intercepts

For horizontal lines, the line doesn't intersect the x-axis unless at infinity, because the line runs parallel to the x-axis. For the y-axis, the line intersects where \(x = 0\). Substituting \(x = 0\) in the equation, \(y = 3\). Thus, the y-intercept is at \((0, 3)\).
04

Draw the Graph

Draw the coordinate plane, mark the y-intercept at \((0, 3)\), and draw a horizontal line through this point parallel to the x-axis. This represents the graph of the equation \(y = 3\).

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Key Concepts

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

Intercepts
Intercepts are key points where a graph crosses the axes on the coordinate plane. These include:
  • **X-Intercept**: The point where the graph intersects the x-axis. At this point, the value for y is 0.
  • **Y-Intercept**: The point where the graph intersects the y-axis. Here, the value for x is 0.
For the equation of a horizontal line, such as \(y = 3\), generally, there is no x-intercept because the line is parallel to the x-axis and never actually touches it unless it's at infinity. However, it always has a clear y-intercept.
To find this, you simply set \(x = 0\) and solve for y. In our equation, when \(x = 0\), \(y \) remains 3. Therefore, the y-intercept of the line \(y = 3\) is the point \((0, 3)\). Identifying intercepts is useful as they provide a starting point for graphing, helping to anchor the graph on the coordinate plane.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which points can be plotted. It is defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is determined by an ordered pair \((x, y)\).

  • **X-Axis**: The horizontal line where y is 0. It's useful for determining x-intercepts.
  • **Y-Axis**: The vertical line where x is 0. It's often used to find y-intercepts.
For graphing, especially linear equations like \(y = 3\), the coordinate plane helps visualize the relationship between variables.
In this case, because \(y\) is always equal to 3, every point on the graph will share the same y-value, forming a perfect horizontal line across the plane. Understanding the coordinate plane enables you to see where lines intersect axes, which aids in complete graph projection.
Horizontal Line
A horizontal line is a straight line that runs left to right on the coordinate plane and does not slant in any direction. The key features of a horizontal line include:
  • **Equation Form**: When an equation is in the form \(y = c\), where \(c\) is a constant, the line is horizontal. For example, \(y = 3\) implies that the y-coordinate is always 3, regardless of the x-coordinates.
  • **Slope**: The slope of a horizontal line is zero because there is no rise over the run between any two points on the line. This means the line does not incline upwards or downwards.
  • **Y-Intercept**: The line will cross the y-axis at \((0, c)\), in our example at \((0, 3)\).
Understanding horizontal lines aids in recognizing patterns and behavior in equations, predicting the graphical pattern when plotted on a coordinate plane.

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Most popular questions from this chapter

Points \(A\) and \(B\) have symmetry with respect to a vertical line \((x=a)\) or a horizontal line \((y=b) .\) Give an equation such as \(x=3\) for the axis of symmetry for: a) \(A(3,-4)\) and \(B(5,-4)\) b) \(\quad A(a, 0)\) and \(B(a,-b)\) c) \(A(7,-4)\) and \(B(-3,-4)\) d) \(A(a, 7)\) and \(B(a,-1)\)

If \(A(2,2), B(7,3),\) and \(C(4, x)\) are the vertices of a right triangle with right angle \(C\), find the value of \(x\).

The real numbers \(a, b, c,\) and \(d\) are positive. Consider the quadrilateral with vertices at \(M(0,0)\) \(N(a, 0), P(a+b, c),\) and \(Q(b, c) .\) Explain why \(M N P Q\) is a parallelogram. CAN'T COPY THE GRAPH

In Exercises 29 to \(34,\) draw the line described. Through \((-2,-5)\) and with \(m=\frac{5}{7}\)

Consider the circle with center \((h, k)\) and radius length \(r .\) If the circle contains the point \((c, d),\) find the slope of the tangent line to the circle at the point \((c, d)\).
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