Practice exercise 4
Without drawing, determine which pair of lines are parallel to each other.
a) [tex]y=3 x-4[/tex] and [tex]3 y-6 x=21[/tex]
b) [tex]y=-\frac{3}{5} x+8[/tex] and [tex]y=\frac{1}{5} x+6[/tex]
c) [tex]y=\frac{3}{4} x-12[/tex] and [tex]12 y=9 x+14[/tex]
d) [tex]2 x=3 y+2[/tex] and another line passing through points [tex]A(6,-5)[/tex] and [tex]B(3,-4)[/tex]
e) [tex]2 y-4 x=3[/tex] and a line passing through points [tex]C(-3,2)[/tex] and [tex]D(-2,4)[/tex].
Find the equations of the lines which are:
Parallel to [tex]y=\frac{1}{2} x-4[/tex] and passes through [tex]P(6,-1)[/tex]
Parallel to [tex]y=-\frac{4}{5} x+3[/tex] and passes through [tex]P(5,-3)[/tex]
Answer (1)
Determine if lines are parallel by comparing their slopes after converting to slope-intercept form.
Calculate the slope between two points using the formula m = x 2 − x 1 y 2 − y 1 .
Use the point-slope form y − y 1 = m ( x − x 1 ) to find the equation of a line parallel to a given line and passing through a specific point.
The pairs of parallel lines are c) and e). The line parallel to y = 2 1 x − 4 and passing through P ( 6 , − 1 ) is y = 2 1 x − 4 . The line parallel to y = − 5 4 x + 3 and passing through P ( 5 , − 3 ) is y = − 5 4 x + 1 .
Explanation
Understanding the Problem We need to determine which pair of lines are parallel and find the equations of lines parallel to given lines and passing through given points. Parallel lines have the same slope. We will convert all equations to slope-intercept form ( y = m x + b ) to compare slopes. Given two points on a line, the slope can be calculated as m = ( y 2 − y 1 ) / ( x 2 − x 1 ) . Find the equation of a line parallel to a given line and passing through a given point using the point-slope form: y − y 1 = m ( x − x 1 ) .
Checking Option A a) y = 3 x − 4 and 3 y − 6 x = 21 ⟹ y = 2 x + 7 . The slopes are 3 and 2, so the lines are not parallel.
Checking Option B b) y = − 5 3 x + 8 and y = 5 1 x + 6 . The slopes are − 5 3 and 5 1 , so the lines are not parallel.
Checking Option C c) y = 4 3 x − 12 and 12 y = 9 x + 14 ⟹ y = 4 3 x + 6 7 . The slopes are 4 3 and 4 3 , so the lines are parallel.
Checking Option D d) 2 x = 3 y + 2 ⟹ y = 3 2 x − 3 2 . The line passing through A ( 6 , − 5 ) and B ( 3 , − 3 ) has slope m = 3 − 6 − 3 − ( − 5 ) = − 3 2 = − 3 2 . The slopes are 3 2 and − 3 2 , so the lines are not parallel.
Checking Option E e) 2 y − 4 x = 3 ⟹ y = 2 x + 2 3 . The line passing through C ( − 3 , 2 ) and D ( − 2 , 4 ) has slope m = − 2 − ( − 3 ) 4 − 2 = 1 2 = 2 . The slopes are 2 and 2, so the lines are parallel.
Finding the First Parallel Line Now, let's find the equation of the line parallel to y = 2 1 x − 4 and passing through P ( 6 , − 1 ) . The slope is 2 1 . Using point-slope form: y − ( − 1 ) = 2 1 ( x − 6 ) ⟹ y + 1 = 2 1 x − 3 ⟹ y = 2 1 x − 4 .
Finding the Second Parallel Line Next, let's find the equation of the line parallel to y = − 5 4 x + 3 and passing through P ( 5 , − 3 ) . The slope is − 5 4 . Using point-slope form: y − ( − 3 ) = − 5 4 ( x − 5 ) ⟹ y + 3 = − 5 4 x + 4 ⟹ y = − 5 4 x + 1 .
Final Answer The pairs of parallel lines are c) and e). The line parallel to y = 2 1 x − 4 and passing through P ( 6 , − 1 ) is y = 2 1 x − 4 . The line parallel to y = − 5 4 x + 3 and passing through P ( 5 , − 3 ) is y = − 5 4 x + 1 .
Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects use parallel lines to create walls, floors, and ceilings that are aligned and stable. The concept of slope is also important in determining the pitch of a roof or the angle of a staircase. By ensuring that lines are parallel and slopes are consistent, architects can create structures that are both aesthetically pleasing and structurally sound. This ensures stability and visual harmony in the design.
