To derive the equation of a line in slope-intercept form, we follow these steps:
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Understand the Slope-Intercept Form: The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept (the point where the line crosses the y-axis).
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Identify Two Points or Slope and a Point: To derive the equation, we typically need either:
- Two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \),
- The slope \( m \) and a point \( (x_1, y_1) \).
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Calculate the Slope (if two points are given): If you have two points, the slope \( m \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
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Use the Point-Slope Formula: If we know a point on the line \( (x_1, y_1) \) and the slope \( m \), we can use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \]
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Convert to Slope-Intercept Form: Once we have the point-slope form, rearranging it will help us find the slope-intercept form: \[ y - y_1 = m(x - x_1) \implies y = mx - mx_1 + y_1 \implies y = mx + (y_1 - mx_1) \] Here, we can identify \( b \) as \( y_1 - mx_1 \).
Example:
Suppose we have two points \( (2, 3) \) and \( (4, 7) \).
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Calculate the Slope: \[ m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \]
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Use Point-Slope Form: Using point \( (2, 3) \): \[ y - 3 = 2(x - 2) \]
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Simplify to Slope-Intercept Form: \[ y - 3 = 2x - 4 \implies y = 2x - 4 + 3 \implies y = 2x - 1 \]
Thus, the equation of the line in slope-intercept form is: \[ y = 2x - 1 \]