**Condition for parallel lines :**

(i) Condition for the lines to be parallel in terms of their slopes.

Let m₁ and m₂ be the slopes of two lines.

If, the two lines are parallel, then their slopes will be equal.

That is,

**m₁ = m₂ **

(ii) Condition for the lines to be parallel in terms of their general form of equations.

Let us consider the general form of equation of a straight line ax + by + c = 0.

If the two lines are parallel, then their general form of equations will differ only in the constant term and they will have the same coefficients of "x" and "y".

That is,

**ax + by + c₁ = 0 **

**ax + by + c₂ = 0 **

(iii) Condition for the lines to be parallel in terms of their slope-intercept form of equations.

Let us consider the slope intercept form of equation of a straight line y = mx + b.

If the two lines are parallel, then their slope-intercept form equations will will differ only in the "y"- intercept.

That is,

**y = mx + b****₁**

** y = mx + b****₂**** **

**(iv) Condition for the lines to be parallel in terms of angle of inclination.**

Let l₁ and l₂ be two lines.

**If the two lines are parallel, the angle between them and the positive side of x-axis will be equal. **

**The figure given below illustrates the above situation.**

**(v) Condition for the lines to be parallel in terms of the perpendicular distance between them. **

**If the two lines are parallel, the perpendicular distance between them will be same at everywhere. **

**The figure given below illustrates the above situation.**

**Problem 1 :**

The slopes of the two lines are 7 and (3k +2). If the two lines are parallel, find the value of "k"

**Solution :**

If the given two lines are parallel, then their slopes are equal.

3k + 2 = 7

3k = 5

**k = 5/3**

**Problem 2 :**

The equations of the two parallel lines are

3x + 2y - 8 = 0

(5k+3) + 2y + 1 = 0

Find the value of "k"

**Solution :**

If the two lines are parallel, then their general form of equations will differ only in the constant term and they will have the same coefficients of "x" and "y".

If the given two lines are parallel, then their slopes are equal.

3k + 2 = 7

3k = 5

**k = 5/3**

**Problem 3 :**

Find the equation of a straight line is passing through (2, 3) and parallel to the line 2x - y + 7 = 0.

**Solution :**

Since the required line is parallel to 2x - y + 7 = 0, the equation of the required line and the equation of the given line 2x - y + 7 = 0 will differ only in the constant term.

Therefore, equation of the required line is 2x -y + k = 0 ------> (1)

The required line is passing through (2, 3).

So, we can plug x = 2 and y = 3 in the equation of the required line.

2(2) - 3 + k = 0

4 - 3 + k = 0

1 + k = 0

k = - 1

**Hence, the equation of the required line is 2x -y - 1 = 0.**

**Problem 4 :**

Verify, whether the two lines 3x + 2y - 7 = 0 and y = -1.5x + 4 are parallel.

**Solution :**

In the equations of the given two lines, the equation of the second line is not in general form.

Let us write the equation of the second line in general form.

y = -1.5x + 4

1.5x + y - 4 = 0

Multiply by 2 on both sides,

3x + 2y - 8 = 0

Now, let us compare the equations of two lines,

3x + 2y - 7 = 0

3x + 2y - 8 = 0

The above two equations differ only in the constant term.

**Hence, the equations of the given two lines are parallel.**

**Problem 5 :**

Verify, whether the two lines 5x + 7y - 1 = 0 and 10x + 14y + 5 = 0 are parallel.

**Solution :**

In the equation of the second line 10x+14y + 5 = 0, the coefficients of "x" and "y" have the common divisor 2.

So, let us divide the second equation by 2

(10x/2) + (14y/2) + (5/2) = (0/2)

5x + 7y + 2.5 = 0

Now, let us compare the equations of two lines,

5x + 7y - 1 = 0

5x + 7y + 2.5 = 0

The above two equations differ only in the constant term.

**Hence, the equations of the given two lines are parallel.**

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