Question

Let line l be represented by the equation 3x − 4y = 1/2 and line m be represented by the equation x − 5 = 2y. Are the lines parallel, perpendicular, or neither? Justify your answer mathematically.

Answer 1

The lines represented by the equations 3x − 4y = 1/2 and x − 5 = 2y are neither parallel nor perpendicular.

Step-by-step explanation:

To determine whether the lines represented by the equations 3x − 4y = 1/2 and x − 5 = 2y are parallel, perpendicular, or neither, we need to compare their slopes. The slope of a line is given by the coefficient of x when the equation is in the form y = mx + b. Let's convert the given equations to slope-intercept form to find their slopes:

1. 3x − 4y = 1/2. Rearrange the equation to isolate y: 4y = 3x - 1/2. Divide both sides by 4: y = (3/4)x - 1/8. So, the slope of line l is 3/4.
2. x − 5 = 2y. Rearrange the equation to isolate y: 2y = x - 5. Divide both sides by 2: y = (1/2)x - 5/2. So, the slope of line m is 1/2.

Since the slopes of the lines are not the same and not negative reciprocals of each other, the lines are neither perpendicular nor parallel.