Both Samuel and Helena are correct, as both slope-intercept form and point-slope form can be used to find the equation of a line given two points. However, the methods differ in how they express the equation.
Slope-Intercept Form
The slope-intercept form of a line is expressed as:
\[ y = mx + b \]
where:
- \( m \) represents the slope of the line.
- \( b \) represents the y-intercept, or the point where the line crosses the y-axis.
To use the slope-intercept form, you first calculate the slope (\( m \)) using the two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Once you have \( m \), you can choose one of the points to find the y-intercept \( b \) by substituting the coordinates into the slope-intercept equation and solving for \( b \).
Point-Slope Form
The point-slope form of a line is expressed as:
\[ y - y_1 = m(x - x_1) \]
where:
- \( (x_1, y_1) \) is a specific point on the line (one of the two points given).
- \( m \) is the slope of the line.
You can calculate the slope \( m \) using the same method as described above. After finding the slope, you can write the equation of the line directly using either of the two points. This form is particularly useful because it allows you to easily see the relationship between the slope and a specific point on the line.
Conclusion
Both forms are equivalent and can be derived from one another. Using either form will lead you to the equation of the line that passes through the two given points. Therefore, whether you use the slope-intercept form or the point-slope form, you will successfully arrive at the equation of the line, making both Samuel and Helena correct in their assertions.
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