Find the equation of the line that is parallel to $y=-x+9$ and contains the point $(7,-13)$.
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Answer (2)
The equation of the line parallel to y = − x + 9 and passing through ( 7 , − 13 ) is found by using the same slope and the point-slope form. The steps involve identifying the slope, substituting the point into the point-slope form, and converting to slope-intercept form. The final equation is: y = − x − 6 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is parallel to a given line and passes through a specific point.
Finding the Slope The given line is y = − x + 9 . We can see that the slope of this line is − 1 . Since parallel lines have the same slope, the line we are looking for also has a slope of − 1 .
Using Point-Slope Form We know the line passes through the point ( 7 , − 13 ) . We can 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.
Substituting Values Plugging in the slope m = − 1 and the point ( 7 , − 13 ) , we get y − ( − 13 ) = − 1 ( x − 7 ) , which simplifies to y + 13 = − x + 7 .
Finding the Equation Now, we rewrite the equation in slope-intercept form, y = m x + b , by subtracting 13 from both sides: y = − x + 7 − 13 , which simplifies to y = − x − 6 .
Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects use parallel lines to ensure walls are aligned and structures are stable. The equation of a line helps in mapping out these structures precisely. If a designer knows one line's equation and needs another parallel line for a specific design element, they can use the principles we applied to find the new line's equation, ensuring the design is both aesthetically pleasing and structurally sound.
The equation of the line parallel to y = − x + 9 that passes through the point ( 7 , − 13 ) is y = − x − 6 . This was determined by identifying the slope, using point-slope form, and rearranging to slope-intercept form.
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