Find the equation of the line that is parallel to $y=-7 x+2$ and contains the point $(-5,32)$.
$y=[?] x+[]$
Answer (2)
The line is parallel to y = − 7 x + 2 , so it has the same slope, which is − 7 .
The equation of the line is of the form y = − 7 x + b .
Substitute the point ( − 5 , 32 ) into the equation to find b : 32 = − 7 ( − 5 ) + b , which simplifies to b = − 3 .
The equation of the line is y = − 7 x − 3 .
Explanation
Understanding the problem We are given a line y = − 7 x + 2 and a point ( − 5 , 32 ) . 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 Since the line we are looking for is parallel to y = − 7 x + 2 , it will have the same slope. The slope of the given line is − 7 . Therefore, the equation of the line we are looking for is of the form y = − 7 x + b , where b is the y-intercept.
Substituting the point To find the value of b , we can substitute the coordinates of the given point ( − 5 , 32 ) into the equation y = − 7 x + b . This gives us 32 = − 7 ( − 5 ) + b .
Solving for the y-intercept Now, we solve for b : 32 = − 7 ( − 5 ) + b 32 = 35 + b b = 32 − 35 b = − 3
Writing the equation Therefore, the equation of the line is y = − 7 x − 3 .
Examples
Understanding parallel lines is crucial in various real-world applications. For instance, consider designing roads or railway tracks; parallel lines ensure that the paths never intersect, maintaining safety and efficiency. Similarly, in architecture, parallel lines are used to create symmetrical and balanced designs, providing structural integrity and aesthetic appeal. In computer graphics, parallel lines are fundamental in creating perspective and depth, enhancing the realism of images and simulations. The ability to find the equation of a line parallel to another is a foundational skill that extends beyond mathematics into practical and creative fields.
The equation of the line that is parallel to y = − 7 x + 2 and passes through the point ( − 5 , 32 ) is y = − 7 x − 3 .
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