Finding the Slope-Intercept and Standard Forms of a Line Given Two Points
When we are given two points on a line, we can find the slope, m, and use it to write the equation of the line in both slope-intercept and standard forms. This process is essential for understanding the relationship between different points and the lines that connect them. For example, given the points (x1, y1) (-3, 4) and (x2, y2) (6, -1), we can derive the equation of the line in various forms.
1. Finding the Slope (m)
The slope m of a line passing through two points can be found using the formula:
m (frac{y_2 - y_1}{x_2 - x_1})
Substituting the given points:
m (frac{-1 - 4}{6 - (-3)} frac{-5}{9})
2. Slope-Intercept Form
The slope-intercept form of a line is given by:
y mx b
First, we know the slope m -(frac{5}{9}). We need to find the y-intercept b. We can use one of the given points, for example, (-3, 4), to solve for b.
Substitute the values into the equation:
4 -(frac{5}{9})(-3) b
4 (frac{15}{9}) b
4 - (frac{15}{9}) b
4 - (frac{5}{3}) b
b (frac{12}{3}) - (frac{5}{3}) (frac{7}{3})
Thus, the equation of the line in slope-intercept form is:
y -(frac{5}{9})x (frac{16}{9})
3. Converting to Standard Form
The standard form of a line is given by:
Ax By C 0
To convert the slope-intercept form to standard form, we start from:
y -(frac{5}{9})x (frac{16}{9})
Multiply every term by 9 to clear the fractions:
9y -5x 16
Adding 5x to both sides:
5x 9y - 16 0
Or, in the more standard form:
5x 9y 16
4. Alternative Approach (Using Point-Slope Form)
Alternatively, we can use the point-slope form of the line, which is given by:
y - y1 m(x - x1)
Using the point (-3, 4) and the slope m -(frac{5}{9}):
y - 4 -(frac{5}{9})(x 3)
Expanding and rearranging for the slope-intercept form:
y - 4 -(frac{5}{9})(em{x} 3)
y - 4 -(frac{5}{9})(em{x} - 5)
y -(frac{5}{9})(em{x} - 20 / 9)
y -(frac{5}{9})(em{x} 4)
5. Conclusion
Both the slope-intercept and standard forms of the line equation can be derived from the given points. The slope-intercept form y -(frac{5}{9})(em{x} 4) shows the line's behavior, while the standard form 5x 9y 16 provides a clear representation of the equation.