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Chapter 9: Problem 29

Determine the intercepts and graph each linear equation. $$3 x+2 y=6$$

Short Answer

Expert verified
x-intercept: (2, 0), y-intercept: (0, 3).

Step by step solution

01

- Determine the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x.\[3x + 2(0) = 6\]So,\[3x = 6\]Therefore,\[x = 2\]The x-intercept is (2, 0).
02

- Determine the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y.\[3(0) + 2y = 6\]So,\[2y = 6\]Therefore,\[y = 3\]The y-intercept is (0, 3).
03

- Plot the intercepts on a graph

Plot the points (2, 0) and (0, 3) on the coordinate plane. These points represent where the line crosses the x-axis and y-axis, respectively.
04

- Draw the line

Using a ruler, draw a straight line through the points (2, 0) and (0, 3). This line represents the equation \(3x + 2y = 6\).

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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** of a linear equation is the point where the line crosses the x-axis. This means the y-coordinate of this point is always zero. To find the x-intercept of the equation \(3x + 2y = 6\), you set \(y = 0\) and solve for \(x\).
Following our example:
  • Set \(y = 0\) in the equation: \(3x + 2(0) = 6\).
  • Simplify to get: \(3x = 6\).
  • Divide by 3: \(x = 2\).
Thus, the x-intercept is \((2, 0)\). This point indicates where the line touches or crosses the x-axis.
y-intercept
The **y-intercept** is where the line crosses the y-axis. This means the x-coordinate of this point is always zero. To find the y-intercept of the equation \(3x + 2y = 6\), you set \(x = 0\) and solve for \(y\).
Following our example:
  • Set \(x = 0\) in the equation: \(3(0) + 2y = 6\).
  • Simplify to get: \(2y = 6\).
  • Divide by 2: \(y = 3\).
Thus, the y-intercept is \((0, 3)\). This point shows where the line touches or crosses the y-axis.
coordinate plane
The **coordinate plane** is a two-dimensional surface on which we can plot points, lines, and curves. It is defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
  • Points on this plane are expressed as \((x, y)\) coordinates.
  • The point \((0, 0)\) is called the origin and is where the x-axis and y-axis intersect.
Using the coordinate plane, we can visualize the x-intercept at \((2, 0)\) and the y-intercept at \((0, 3)\). These points will help us draw the line that represents our equation.
graphing
**Graphing** a linear equation involves plotting points on the coordinate plane and drawing a line through these points. For \(3x + 2y = 6\), we have determined the intercepts: \((2, 0)\) and \((0, 3)\).
Here's the process:
  • Plot the x-intercept \((2, 0)\) on the coordinate plane.
  • Plot the y-intercept \((0, 3)\) on the coordinate plane.
  • Use a ruler to draw a straight line that passes through these two points.
This line represents the equation \(3x + 2y = 6\). Every point on this line is a solution to the equation.

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Most popular questions from this chapter

In Exercises \(33-38\), determine whether the lines with the given equations are parallel, perpendicular, or neither. \(2 x+1=0\) \(-3 y-4=0\)

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Find the missing coordinate so that each ordered pair is a solution to the equation. $$3 y+1=0$$ (a) \((0, ?) ;\) (b) \((?, 0) ;\) (c) \((3, ?) ; \quad\) (d) \((?,-1)\)

Plot the points corresponding to the ordered pairs \(A(1,4), B(4,-2)\) \(C(-3,2), D(-3,0), E(3,0), F(0,0), G(-3,-3),\) and \(H(0,-2)\)

Suppose the origin is the midpoint of a segment and one endpoint of the segment is \((x, y) .\) Find the coordinates of the other endpoint.
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