Determine a if the lines are parallel: x+y=1 and 3x-ay=2
If lines are parallel their slopes are equal.
Since both the equations are written in standard form (Ax + By = C) we can determine the slope by using the formula: -B/A.
1. First equation: x + y = 1: A = 1, B = 1, so -B/A = -(1)/(1) = -1
2. Second equation: 3x - ay = 2: A = 3, B = -a, so -B/A = -(-a)/(3) = a/3.
Again, if lines are parallel their slopes are equal, so we can set the slopes equal to each other in order to solve for a.
3. -1 = a/3 To solve for a multiply both sides by 3.
4. (-1)(3) = (a/3)(3) Simplify.
5. -3 = a
a = -3
To determine if x+y =1 and 3x-3ay =2 are parallel.
Two lines are parallel if they have the same slope.
We know that the the equation of any line in the slope intercept form is y = mx+c, wher m is the slope.
So we convert the given lines into the slope intercept form as below;
x+y = 1.
Subtract x :
y = -x+1 which has a slope -1......(1).
So y = -x+1 is the slope intercept form of x+y = 1.
Consider the second line 3x-ay =2. Subtract 3x from both sides:
-ay = -3x +2.
Divide by -a.
y = -3x/(-a) +2/(-a)
y = (3/a)x +(-2/a)...........(2)
This line has a slope of 3/a.
If the two line are parallel , the their slopes should be equal;
Therefore from (1) and (2),
3/a = -1
Multiply by a:
3 = -a. Or
So if a = -3 , then the given two lines would be parallel.
a = -3.
We have to determine a if the two lines x+y=1 and 3x-ay=2 are parallel.
Now for a line ax + by +c =0 , the slope is given as -a/b
For the line x+y=1
=> x + y -1 =0
=> slope is -1
For the line 3x - ay =2
=> 3x -ay -2 =0
=> the slope is -3 / -a = 3/a
Now this has to be equal to -1 as the two lines are parallel.
Hence 3/a = -1
=> a = -3
Therefore the required value of a is -3
If the 2 given lines are parallel, then the values of their slopes are equal.
The given equations are x+y-1=0 and 3x-ay-2=0, so we'll have to put them in the standard from, which is y = mx+n.
We'll begin with x+y=1.
We'll isolate y to the left side:
y = -x+1
So, the slope can be easily determined as m1 = -1.
That means that the second slope has the same value, m2 = -1.
We'll put the other equation into the standard form, isolating y to the left side. For this reason we'll subtract 3x both sides:
-ay = -3x + 2
We'll divide by -a:
y = 3x/a - 2/a
The slope m2 = 3/a
But m2 = -1, so 3/a = -1
The line is now determined, having as equation: 3x+3y-2=0