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Equation of a Line

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A straight line is a curve, where every point on the line segment joins any two points lying on it. So, every first degree equation in x and y represents a straight line. There is always a fixed relationship between the x and y co-ordinates of any points on the line. For example: x + y = 0, 4x + y = 0, y = 0.

A straight line, when seen in a two-dimensional space, it contains a unique property, where the ratio of the difference in y- coordinates of any points on the line to the difference of their x-coordinates is always the same. That is, how the slope of the straight line remains constant and can be calculated from any two given points on the line.

Standard Form of Equation of a Line

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The Standard form of equation of line is defined as Ax + By = C, where A $\neq$ 0 and B $\neq$ 0.
A and B are the coefficients of x and y respectively. The ordered pair r = (x, y) can be defined as a point.

Standard form of equation is mainly used under the following circumstances:
  • When we want to graph a line
  • When we want to know the y-intercept of the line
  • When we want to know the x-intercept 
Standard form of Equation of a Line

Equation of Line Formula

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The general formula of equation of line passing through a point ($x_1,y_1$) and having slope m is given as

Equation of line formula
Here, 'm' is the slope of the line
x1 is the Co-ordinate of x-axis
y1 is the Co-ordinate of y-axis

Here are few solved example on the equation of line:

Solved Examples

Question 1: Solve the equation of the lines whose slope is 7 and one of the point is (4, 5).
Solution:
Given: m = 7

(x1, y1) <-> (4, 5)

Using the formula of the equation of line, we get

$y-y_{1}$ = $m(x-x_{1})$

y - 5 = 7(x - 4)

y - 5 = 7x - 28

7x - y - 28 + 5 = 0

7x - y - 23 = 0

This is the required equation of line.

Question 2: Calculate the equation of the line for which the slope is 3 and passes through the point (8, 9).
Solution:
Given: m = 3

(x1, y1) <-> (8, 9)

Using the formula of equation of the line, we get

$y-y_{1}$ = $m(x-x_{1})$

(y - 9) = 3( x - 8 )

y - 9 = 3x - 24

3x - y - 24 + 9 = 0

3x - y - 15 = 0

Slope Intercept Equation of a Line

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The slope intercept form of a line is defined as a straight line on the co-ordinate plane that can be explained by the equation

Slope Intercept Equation of a LineWhere, m is the slope of the line and b is the y-intercept and x and y are the co-ordinates of any point on the line.

Slope Intercept Equation of Lines

Slope (m) is the 'steepness' of the line and 'b' is the intercept, that is the point where the line crosses the y-axis.

This equation of the line is mainly used in the following circumstances:
  • To define a particular line in an accurate way.
  • To locate the points on the line.
Given below are the few solved problems based on the slope intercept equation of a line.

Solved Examples

Question 1: The slope of a line is 7 and the y-intercept is 4. Calculate the equation of the line.
Solution:
Step 1: Given:

m = 7 and b = 4

Using the formula y = mx + b, we get 

y = 7x + 4

7x - y + 4 = 0

$\therefore$ The equation of the line is 7x - y + 4 = 0

Step 2: Given:

m = 7 and b = 4

Using the formula y = mx + b, we get 

y = 7x + 4

7x - y + 4 = 0

$\therefore$ The equation of the line is 7x - y + 4 = 0

Question 2: Calculate the equation of the line of the slope 0.2 and y-intercept 7.
Solution:
Given:

m = 0.2 and c = 7

Using the formula y = mx + c, we get

y = 0.2 (x) + 7

y = $\frac{x}{5}$ + 7

5y = x + 35

x - 5y + 35 = 0

$\therefore$ The equation of the line is x - 5y + 35 = 0.

Parametric Equation of the Line

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Parametric equations are the one where the Cartesian co-ordinate of a curve or a surface are represented as the function of the same variable.

The parametric equation in the xy - plane is explained as x = x(t) and y = y(t)

x and y denote the co-ordinate of the graph of a curve in the plane. The parametric equation of the line is defined as the line on a co-ordinate plane given with the point P1 (x1, y1) and the direction vector s then, the position vector r of any point (x, y) of the line.

r = r1 + t.s, - $\infty$ < t < + $\infty$

and where, r1 = x1i + y1j and s = xsi + ysj, represents the vector equation of the line.
Therefore, any point of line can be reached by the radius vector

r = xi + yj = (x1 + xst)i + (y1 + yst)j

$\therefore$ the scalar quantity 't' can take any real value from - $\infty$ to + $\infty$. By rewriting the scalar components of the above equation, we get the parametric equation of the line as

x = x1 + xs.t
y = y1 + ys.t