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If a point lies on a line, then it must satisfy the equation of the line.

Consider a horizontal line L that intersects the positive side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line L at any position is 'a'.

Equation of line *l*: *y* = *a*

Consider another horizontal line S that intersects the negative side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line S at any position is minus 'a'.

Equation of line *s*: *y* = -*a*

Consider a vertical line L that intersects the positive side of the X-axis at a distance 'b' from the Y-axis.

The x coordinate of every point on line L at any position is 'b'.

Equation of line *l*: *x* = *b*

Consider another vertical line S that intersects the negative side of the X-axis at a distance 'c' from the Y-axis.

The x coordinate of every point on line S at any position is minus 'c'.

Equation of line *s*: *y* = -*c*

**Point-Slope Form**

Consider a non-vertical line L passing through a given point P(x_{1}, y_{1}).

Let m be the slope of line L.

Let A(x, y) be any arbitrary point on line L.

The slope of a line passing through (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) is (*y*_{2} - *y*_{1})/(*x*_{2} - *x*_{1}).

Here, slope (*m)* = (*y* - *y*_{1})/(*x* - *x*_{1})

â‡’ *y* - *y*_{1} = *m* (*x* - *x*_{1}) â€¦.(1)

Equation 1 represents the point-slope form of the equation of a line passing through a given point (x_{1}, y_{1}) and with slope m.

**Two-Point Form**

Consider a non-vertical line L passing through two given points P and Q.

Let A be any arbitrary point on line L.

Three given points A, B and C are collinear if: Slope of AB = Slope of BC

Thus, A, P and Q are collinear if: Slope of PA = Slope of PQ

Slope of PA = (*y* - *y*_{1})/ (*x* - *x*_{1})

Slope of PQ = (*y*_{2} - *y*_{1})/ (*x*_{2} - *x*_{1})

â‡’ (*y* - *y*_{1})/ (*x* - *x*_{1}) = (*y*_{2} - *y*_{1})/ (*x*_{2} - *x*_{1})

This equation represents the two-point form of the equation of a line passing through the given points, P and Q.

If a point lies on a line, then it must satisfy the equation of the line.

Consider a horizontal line L that intersects the positive side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line L at any position is 'a'.

Equation of line *l*: *y* = *a*

Consider another horizontal line S that intersects the negative side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line S at any position is minus 'a'.

Equation of line *s*: *y* = -*a*

Consider a vertical line L that intersects the positive side of the X-axis at a distance 'b' from the Y-axis.

The x coordinate of every point on line L at any position is 'b'.

Equation of line *l*: *x* = *b*

Consider another vertical line S that intersects the negative side of the X-axis at a distance 'c' from the Y-axis.

The x coordinate of every point on line S at any position is minus 'c'.

Equation of line *s*: *y* = -*c*

**Point-Slope Form**

Consider a non-vertical line L passing through a given point P(x_{1}, y_{1}).

Let m be the slope of line L.

Let A(x, y) be any arbitrary point on line L.

The slope of a line passing through (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) is (*y*_{2} - *y*_{1})/(*x*_{2} - *x*_{1}).

Here, slope (*m)* = (*y* - *y*_{1})/(*x* - *x*_{1})

â‡’ *y* - *y*_{1} = *m* (*x* - *x*_{1}) â€¦.(1)

Equation 1 represents the point-slope form of the equation of a line passing through a given point (x_{1}, y_{1}) and with slope m.

**Two-Point Form**

Consider a non-vertical line L passing through two given points P and Q.

Let A be any arbitrary point on line L.

Three given points A, B and C are collinear if: Slope of AB = Slope of BC

Thus, A, P and Q are collinear if: Slope of PA = Slope of PQ

Slope of PA = (*y* - *y*_{1})/ (*x* - *x*_{1})

Slope of PQ = (*y*_{2} - *y*_{1})/ (*x*_{2} - *x*_{1})

â‡’ (*y* - *y*_{1})/ (*x* - *x*_{1}) = (*y*_{2} - *y*_{1})/ (*x*_{2} - *x*_{1})

This equation represents the two-point form of the equation of a line passing through the given points, P and Q.