For which value of $a$ are the graphs of $-4=3 x+6 y$ and $a x-8 y=12$ parallel?
Answer (1)
Rewrite the first equation in slope-intercept form to find its slope: y = − 2 1 x − 3 2 .
Rewrite the second equation in slope-intercept form to find its slope: y = 8 a x − 2 3 .
Set the slopes equal to each other since the lines are parallel: − 2 1 = 8 a .
Solve for a : a = − 4 .
− 4
Explanation
Understanding Parallel Lines We are given two equations: − 4 = 3 x + 6 y and a x − 8 y = 12 . We need to find the value of a for which the graphs of these equations are parallel. Two lines are parallel if and only if they have the same slope and different y-intercepts.
Finding the Slope of the First Line First, we rewrite the first equation in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. We have: − 4 = 3 x + 6 y Subtract 3 x from both sides: − 3 x − 4 = 6 y Divide by 6: y = 6 − 3 x − 4 = − 2 1 x − 3 2 So the slope of the first line is m 1 = − 2 1 .
Finding the Slope of the Second Line Next, we rewrite the second equation in slope-intercept form: a x − 8 y = 12 Subtract a x from both sides: − 8 y = − a x + 12 Divide by -8: y = − 8 − a x + 12 = 8 a x − 2 3 So the slope of the second line is m 2 = 8 a .
Equating the Slopes Since the lines are parallel, their slopes must be equal. Therefore, we set m 1 = m 2 :
− 2 1 = 8 a Multiply both sides by 8: a = − 2 1 × 8 = − 4
Final Answer Thus, the value of a for which the lines are parallel is a = − 4 .
Examples
Understanding parallel lines is crucial in various real-world applications, such as designing roads and buildings. For instance, architects use the concept of parallel lines to ensure that walls and floors are aligned correctly, providing structural stability and aesthetic appeal. Similarly, civil engineers apply this principle when constructing parallel lanes on highways, ensuring smooth traffic flow and minimizing accidents. By grasping the mathematical properties of parallel lines, professionals can create safer and more efficient designs in numerous fields.
