Please show me how to solve:
For what value of k are the graphs of 12y = -3x + 8 and 6y =kx - 5 parallel? For what value of k are they perpendicular?
if lines are parallel, they have the same slope
so, the first step is to convert each equation to the slope-intercept form:
y = -1/4 x + 2/3
y = k/6 x - 5/6
If the slopes are the same, then
-1/4 = k/6
k = -3/2
If the lines are perpendicular, their slopes are negative reciprocals; or, their product is -1.
-1/4 * k/6 = -1
-k/24 = -1
k = 24
To determine the value of k for which the graphs of the equations 12y = -3x + 8 and 6y = kx - 5 are parallel, we need to compare their slopes. The slopes of two parallel lines are equal.
In both equations, we can rewrite them in slope-intercept form (y = mx + b) to easily determine their slopes.
First, let's rewrite the equation 12y = -3x + 8 in slope-intercept form:
Divide both sides of the equation by 12: y = (-1/4)x + 2/3.
From this equation, we can see that its slope is -1/4.
Now, let's rewrite the equation 6y = kx - 5 in slope-intercept form:
Divide both sides of the equation by 6: y = (k/6)x - 5/6.
From this equation, we can see that its slope is k/6.
To find the value of k where these slopes are equal (parallel lines), we set the slopes equal to each other and solve for k:
-1/4 = k/6.
To solve this equation, you can cross multiply: -6 = 4k.
Then, divide both sides by 4 to isolate k: k = -6/4, or simplifying further, k = -3/2.
Therefore, for the graphs to be parallel, the value of k is -3/2.
Next, let's find the value of k for which the graphs are perpendicular.
Two lines are perpendicular when the product of their slopes is -1.
The slope of the first equation is -1/4, and we can rewrite the other equation as y = (k/6)x - 5/6.
Now we set the product of the slopes equal to -1 and solve for k:
(-1/4) * (k/6) = -1.
To solve this equation, you can cross multiply: -k/24 = -1.
Then, multiply both sides by -24 to isolate k: k = 24.
Therefore, for the graphs to be perpendicular, the value of k is 24.
To summarize:
- For the graphs to be parallel, the value of k is -3/2.
- For the graphs to be perpendicular, the value of k is 24.