Find an equation for the line with the given property.
(a) It passes through the point $(5,-7)$, and is parallel to the line $2 x+y-10=0$
Answer (1)
Find the slope of the given line 2 x + y − 10 = 0 , which is − 2 .
Use the same slope, − 2 , for the parallel line.
Apply the point-slope form with the point ( 5 , − 7 ) : y − ( − 7 ) = − 2 ( x − 5 ) .
Simplify to get the equation in slope-intercept form: y = − 2 x + 3 .
Explanation
Understanding the Problem We are given a point ( 5 , − 7 ) and a line 2 x + y − 10 = 0 . We need to find the equation of a line that passes through the given point and is parallel to the given line.
Finding the Slope of the Given Line First, we need to find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Starting with 2 x + y − 10 = 0 , we can isolate y to get y = − 2 x + 10 . Therefore, the slope of the given line is − 2 .
Determining the Slope of the Parallel Line Since the line we want to find is parallel to the given line, it must have the same slope. So, the slope of our desired line is also − 2 .
Using the Point-Slope Form Now we can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope. We have the point ( 5 , − 7 ) and the slope − 2 . Plugging these values into the point-slope form, we get: y − ( − 7 ) = − 2 ( x − 5 ) .
Simplifying the Equation Simplify the equation: y + 7 = − 2 x + 10 . Now, we can rewrite the equation in slope-intercept form by subtracting 7 from both sides: y = − 2 x + 10 − 7 , which simplifies to y = − 2 x + 3 . Alternatively, we can write the equation in standard form A x + B y = C . Starting from y = − 2 x + 3 , add 2 x to both sides to get 2 x + y = 3 .
Final Answer The equation of the line that passes through the point ( 5 , − 7 ) and is parallel to the line 2 x + y − 10 = 0 is y = − 2 x + 3 in slope-intercept form or 2 x + y = 3 in standard form.
Examples
Understanding parallel lines is crucial in architecture and design. For instance, when designing a building, architects ensure that walls are parallel to each other for structural stability and aesthetic appeal. If a wall needs to pass through a specific point and be parallel to another wall, the principles used in this problem can be applied to determine the equation representing the new wall's alignment, ensuring it meets the design requirements.
