Find the equation of the line that is parallel to $y=x-3$ and contains the point $(3,-2)$.
$y=[?] x+[]$
Answer (1)
Determine the slope of the given line y = x − 3 , which is 1 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( 3 , − 2 ) and slope 1 .
Simplify the equation to slope-intercept form: y = x − 5 .
The equation of the line is y = x − 5 .
Explanation
Understanding the Problem We are given a line y = x − 3 and a point ( 3 , − 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 given line y = x − 3 is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In this case, the slope of the given line is m = 1 . Therefore, the slope of the line we are looking for is also 1 .
Using Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. We are given the point ( 3 , − 2 ) , so x 1 = 3 and y 1 = − 2 . We found that the slope is m = 1 . Plugging these values into the point-slope form, we get:
y − ( − 2 ) = 1 ( x − 3 )
Simplifying to Slope-Intercept Form Now we simplify the equation and rewrite it in slope-intercept form, y = m x + b :
y + 2 = x − 3
Subtract 2 from both sides:
y = x − 3 − 2
y = x − 5
Final Answer The equation of the line is y = x − 5 . Therefore, the slope is 1 and the y-intercept is − 5 .
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. If you're designing a rectangular room and one wall follows the equation y = x − 3 , a parallel wall might follow y = x − 5 , maintaining a consistent aesthetic and structural integrity.
