# Determine a if the lines are parallel: x+y=1 and 3x-ay=2

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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:

3x-ay=2

-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

**a= -3**

**The line is now determined, having as equation: 3x+3y-2=0**