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Magazine Chapter 2: Problem 89
Explain how to graph a line using the intercept method.
Short Answer
Expert verified
Find the intercepts and draw a line through them.
Step by step solution
01
Understand the Line Equation
The standard form of a line is given by the equation \( Ax + By = C \). To graph a line using the intercept method, start by understanding this form, where \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables representing coordinates on the Cartesian plane.
02
Find the X-Intercept
The x-intercept is the point where the line crosses the x-axis. At this point, the value of \( y \) is 0. Substitute \( y = 0 \) into the equation \( Ax + By = C \) to solve for \( x \). This will give you the x-coordinate of the x-intercept. For example, if the equation is \( 2x + 3y = 6 \), substituting \( y = 0 \) gives \( 2x = 6 \), or \( x = 3 \). Therefore, the x-intercept is \( (3, 0) \).
03
Find the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. At this point, the value of \( x \) is 0. Substitute \( x = 0 \) into the equation \( Ax + By = C \) to solve for \( y \). This will give you the y-coordinate of the y-intercept. Using the same example \( 2x + 3y = 6 \), substituting \( x = 0 \) gives \( 3y = 6 \), or \( y = 2 \). Therefore, the y-intercept is \( (0, 2) \).
04
Plot the Intercepts
On a graph, plot the x-intercept and y-intercept points that you have found. Using the example, plot the points \( (3, 0) \) and \( (0, 2) \) on the Cartesian plane.
05
Draw the Line
Draw a straight line through the two intercepts you plotted. This line represents the graph of the equation \( 2x + 3y = 6 \). Extend the line across the graph to indicate that it continues indefinitely.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Intercept Method
The intercept method is a straightforward technique for graphing linear equations. It involves identifying two specific points, the x-intercept and the y-intercept, where a line crosses the axes on a graph. By finding these points, you can accurately plot the line. This method is efficient because it requires only simple substitutions and calculations.
Here鈥檚 a quick way to remember the steps:
Here鈥檚 a quick way to remember the steps:
- Find the x-intercept: set \( y = 0 \) and solve for \( x \).
- Find the y-intercept: set \( x = 0 \) and solve for \( y \).
- Plot the intercepts on the Cartesian plane.
- Draw the line through the plotted points.
X-Intercept
The x-intercept of a line is the point where the graph of the equation crosses the x-axis. It is significant because it tells you where the line touches the horizontal axis.
To find the x-intercept:
To find the x-intercept:
- Set \( y = 0 \) in the equation.
- Solve for \( x \) to get the x-coordinate of the intercept.
Y-Intercept
The y-intercept of a line is another key point on a graph, marking where the line intersects the y-axis. This point helps you determine how the line behaves in relation to the vertical axis.
Follow these steps to find the y-intercept:
Follow these steps to find the y-intercept:
- Set \( x = 0 \) in the equation.
- Solve for \( y \) to get the y-coordinate of the intercept.
Cartesian Plane
The Cartesian plane, or coordinate plane, is the canvas on which we graph our equations. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, allowing us to plot points through coordinates \( (x, y) \).
Understanding the Cartesian plane is crucial for graphing:
Understanding the Cartesian plane is crucial for graphing:
- The x-axis runs left to right, and the y-axis runs up and down.
- Coordinates are written in the form \( (x, y) \), where \( x \) shows the position relative to the vertical axis and \( y \) shows the position relative to the horizontal axis.
- You can graph lines, curves, points, and shapes by marking points given as coordinates.
