Determine the slope-intercept form of the equation of the line parallel to [tex]y=-\frac{4}{3} x+11[/tex] that passes through the point (-6,2).
[tex]y=\square x+\square[/tex]
Answer (2)
The slope of the given line is − 3 4 . Using the point-slope form, we find that the equation of the parallel line is y = − 3 4 x − 6 .
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Determine the slope of the given line: m = − 3 4 .
Use the point-slope form with the given point ( − 6 , 2 ) : y − 2 = − 3 4 ( x + 6 ) .
Convert to slope-intercept form: y = − 3 4 x − 6 .
The equation of the parallel line is y = − 3 4 x − 6 .
Explanation
Understanding the Problem We are given a line with the equation y = − 3 4 x + 11 and a point ( − 6 , 2 ) . We need to find the equation of a line that is parallel to the given line and passes through the given point.
Finding the Slope Parallel lines have the same slope. The slope of the given line is − 3 4 . Therefore, the slope of the line we want to find is also − 3 4 .
Using Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point. In our case, m = − 3 4 and ( x 1 , y 1 ) = ( − 6 , 2 ) . Substituting these values into the point-slope form, we get:
y − 2 = − 3 4 ( x − ( − 6 ))
Converting to Slope-Intercept Form Now, we simplify the equation and convert it to slope-intercept form ( y = m x + b ):
y − 2 = − 3 4 ( x + 6 )
y − 2 = − 3 4 x − 3 4 ( 6 )
y − 2 = − 3 4 x − 8
y = − 3 4 x − 8 + 2
y = − 3 4 x − 6
Final Answer The equation of the line parallel to y = − 3 4 x + 11 that passes through the point ( − 6 , 2 ) is y = − 3 4 x − 6 .
Examples
Understanding parallel lines is crucial in various real-world applications. For instance, consider designing a road with multiple lanes. The lanes must be parallel to ensure smooth traffic flow and prevent accidents. Similarly, in architecture, parallel lines are used to create symmetrical and aesthetically pleasing structures. Knowing how to determine the equation of a line parallel to another helps in these scenarios by ensuring that lines maintain a constant distance from each other, providing stability and visual harmony.
