For which value of $a$ are the graphs of $-4=3 x+6 y$ and $a x-8 y=12$ parallel?
A. $a=-4$
B. $a=-\frac{1}{2}$
C. $a=\frac{1}{4}$
D. $a=2
Answer (2)
The value of a that makes the graphs of − 4 = 3 x + 6 y and a x − 8 y = 12 parallel is a = − 4 . This is found by ensuring both lines have the same slope but different y-intercepts. Therefore, the answer is A. a = − 4 .
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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 express its slope in terms of a : y = 8 a x − 2 3 .
Set the slopes of the two lines equal to each other and solve for a : 8 a = − 2 1 ⇒ a = − 4 .
Check that the y-intercepts are different for the two lines with the found value of a : − 3 2 = − 2 3 . The final answer is − 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 they have the same slope but different y-intercepts.
Finding Slope and Intercept of the First Line First, let's 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: 3 x + 6 y = − 4 6 y = − 3 x − 4 y = 6 − 3 x − 6 4 y = − 2 1 x − 3 2 So, the slope of the first line is − 2 1 and the y-intercept is − 3 2 .
Finding Slope and Intercept of the Second Line Now, let's rewrite the second equation in slope-intercept form: a x − 8 y = 12 − 8 y = − a x + 12 y = − 8 − a x + − 8 12 y = 8 a x − 2 3 So, the slope of the second line is 8 a and the y-intercept is − 2 3 .
Equating Slopes For the lines to be parallel, their slopes must be equal. Therefore, we set the slopes equal to each other: 8 a = − 2 1 a = 8 × − 2 1 a = − 4
Checking Y-Intercepts Now we need to check if the y-intercepts are different. The y-intercept of the first line is − 3 2 , and the y-intercept of the second line is − 2 3 . Since − 3 2 = − 2 3 , the y-intercepts are different.
Final Answer Therefore, the value of a for which the lines are parallel is − 4 .
Examples
Understanding parallel lines is crucial in various real-world applications. For instance, consider designing city streets; parallel streets ensure efficient traffic flow and prevent intersections that could cause congestion. In architecture, parallel lines are used to create aesthetically pleasing and structurally sound buildings. Moreover, in computer graphics, parallel lines are fundamental for creating realistic perspectives and rendering objects accurately. The ability to determine when lines are parallel, as demonstrated in this problem, is a valuable skill in many fields.
