To graph the line given by the equation \( y = x + 3 \), you can follow these steps using the slope and y-intercept:
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Identify the y-intercept: In the equation \( y = x + 3 \), the y-intercept is \( 3 \). This is the point where the line crosses the y-axis. You can plot this point on the graph:
- Point: \( (0, 3) \)
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Identify the slope: The slope of the line is \( 1 \) (the coefficient of \( x \)). This means that for every 1 unit you move up on the y-axis, you move 1 unit to the right on the x-axis. Thus:
- From the y-intercept \( (0, 3) \), move up 1 unit to \( (0, 4) \) and 1 unit to the right to \( (1, 4) \).
- Plot this second point: \( (1, 4) \)
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Plot a few more points: You can continue using the slope to find additional points:
- From \( (1, 4) \), move up 1 unit and 1 unit to the right to get to \( (2, 5) \) and plot this point.
- You can also move backwards from the y-intercept: From \( (0, 3) \), move down 1 unit to \( (0, 2) \) and 1 unit to the left to get to \( (-1, 2) \) and plot this point.
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Draw the line: After plotting at least two points (the y-intercept and another point), draw a straight line through all the points you have plotted. This line represents the equation \( y = x + 3 \).
The line will have a positive slope and will pass through the points \( (0, 3) \), \( (1, 4) \), and \( (2, 5) \), and continue infinitely in both directions.
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