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Chapter 3: Problem 20

Graph each linear equation by finding and plotting its intercepts. See Examples 6 through 8 . $$ 2 x+3 y=6 $$

Short Answer

Expert verified
The graph is a straight line through points (3, 0) and (0, 2).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 2x + 3y = 6 \).\[2x + 3(0) = 6 \2x = 6 \x = \frac{6}{2} \x = 3\]The x-intercept is \( (3, 0) \).
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 2x + 3y = 6 \).\[2(0) + 3y = 6 \3y = 6 \y = \frac{6}{3} \y = 2\]The y-intercept is \( (0, 2) \).
03

Plot the intercepts

On a coordinate plane, plot the intercepts found: \( (3, 0) \) and \( (0, 2) \).
04

Draw the line through the intercepts

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

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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 is where the graph of an equation crosses the x-axis on the coordinate plane. To find this point, you set the value of y to zero in the equation and solve for x. This is because all points on the x-axis have a y-coordinate of zero.

  • In the provided exercise, the equation is \(2x + 3y = 6\).
  • Setting \(y = 0\), you get \(2x + 3(0) = 6\), simplifying to \(2x = 6\).
  • Solving for x, you divide both sides by 2, giving you \(x = 3\).
This results in the x-intercept being the point \((3, 0)\). By understanding how to locate the x-intercept, you learn where the line will intersect the x-axis, which is crucial when drawing the graph of the equation.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. To find it, set the x variable to zero in the equation and solve for y, since every point on the y-axis has an x-coordinate of zero.

  • For the equation \(2x + 3y = 6\), set \(x = 0\).
  • This gives \(2(0) + 3y = 6\), simplifying to \(3y = 6\).
  • Divide both sides by 3 to find \(y = 2\).
Thus, the y-intercept is \((0, 2)\). Understanding the y-intercept helps in graphing linear equations, as it gives a clear point on the graph where the line will pass through the y-axis.
Coordinate Plane
A coordinate plane is a two-dimensional plane formed by two perpendicular number lines: the x-axis running horizontally and the y-axis running vertically. These axes divide the plane into four quadrants, each providing a unique area where positive and negative values intersect.

  • The point of intersection of the x-axis and y-axis is known as the origin \((0,0)\).
  • Using the coordinate plane allows you to plot points such as the intercepts you've found, \((3,0)\) and \((0, 2)\).
  • Once these intercepts are plotted, you can draw a line through them to represent the equation \(2x + 3y = 6\).
The coordinate plane is an essential tool for graphing linear equations efficiently. By understanding the layout and use of a coordinate plane, you can visualize mathematical relationships effectively.

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Most popular questions from this chapter

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