Introduction to Finding the Equation of a Line
Understanding the equation of a line often involves identifying its slope and intercepts. This article will walk through the process of determining the equation of a line that makes a 135° angle with the positive x-axis and cuts an intercept of -3 from the negative y-axis.
Understanding the Angle and Intercept
A line can be described by its slope and y-intercept. In this case, the problem specifies that the line makes an angle of 135° with the positive x-axis and has a y-intercept of -3. The angle provides the slope of the line, which is critical for determining the line's equation.
Calculating the Slope
The slope ( m ) of the line can be calculated using trigonometry. Specifically, the angle between the line and the positive x-axis is 135°. The tangent of this angle can be calculated as:
( m tan(135°) )
Since tangent is periodic and (tan(135°) tan(180° - 45°)), and knowing that (tan(180° - theta) -tan(theta)), we have:
( m -tan(45°) -1 )
Hence, the slope ( m ) of the line is (-1).
Formulating the Equation of the Line
The general form of the line's equation is given by the y-intercept form ( y mx c ), where:
m is the slope of the line, c is the y-intercept.Given:
( m -1 ) ( c -3 )The equation of the line is:
( y -x - 3 )
Conclusion
We have determined the equation of the line by finding its slope and y-intercept. The final equation of the line, having a slope of (-1) and a y-intercept of (-3), is:
( y 3 -x )
This can be rewritten as:
( xy 3 0 )
Thus, the equation of the line is:
( xy 3 0 )