What is the equation of the line that is parallel to the given line and passes through the point $(2,3)$?
$x+2 y=4$
$x+2 y=8$
$2 x+y=4$
$2 x+y=8$
Answer (1)
Find the slope of the given line by rewriting it in slope-intercept form: y = − 2 1 x + 2 , so the slope is − 2 1 .
Use the point-slope form of a line with the point ( 2 , 3 ) and the slope − 2 1 : y − 3 = − 2 1 ( x − 2 ) .
Rewrite the equation in standard form: x + 2 y = 8 .
The equation of the line is x + 2 y = 8 .
Explanation
Understanding the Problem We are given the line x + 2 y = 4 and the point ( 2 , 3 ) . We want to find the equation of the line that is parallel to the given line and passes through the given point.
Finding the Slope 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.
Slope of the Given Line The given equation is x + 2 y = 4 . Subtracting x from both sides, we get 2 y = − x + 4 . Dividing both sides by 2, we get y = − 2 1 x + 2 . Therefore, the slope of the given line is − 2 1 .
Slope of the Parallel Line Since we want to find a line that is parallel to the given line, the slope of the parallel line will be the same as the slope of the given line, which is − 2 1 .
Using 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. In this case, ( x 1 , y 1 ) = ( 2 , 3 ) and m = − 2 1 .
Finding the Equation Plugging in the values, we get y − 3 = − 2 1 ( x − 2 ) . Multiplying both sides by 2, we get 2 ( y − 3 ) = − ( x − 2 ) . Expanding, we get 2 y − 6 = − x + 2 . Adding x to both sides, we get x + 2 y − 6 = 2 . Adding 6 to both sides, we get x + 2 y = 8 .
Final Answer Therefore, the equation of the line that is parallel to the given line and passes through the point ( 2 , 3 ) is x + 2 y = 8 .
Examples
Imagine you're designing a ramp for a building. You know the slope you need for the ramp to be accessible, and you have a specific point where the ramp needs to start. Using the concept of parallel lines, you can determine the exact equation for the ramp's surface, ensuring it meets the required slope and starts at the correct location. This ensures the ramp is both accessible and properly aligned with the building's entrance.
