Equation of a Straight Line

We will discuss here about the meaning of equation of a straight line.

Let the straight line be PQ which passes through the origin (0, 0) and inclined at 45° with the positive direction of the x-axis. Let the points on the line PQ are (x$$_{1}$$, y$$_{1}$$), (x$$_{2}$$, y$$_{2}$$), (x$$_{3}$$, y$$_{3}$$), etc.,

According to the definition of coordinates $$\frac{y_{1}}{x_{1}}$$ = tan 45° = $$\frac{y_{2}}{x_{2}}$$ = $$\frac{y_{3}}{x_{3}}$$ = etc.,

Therefore, y$$_{1}$$ = x$$_{1}$$, y$$_{2}$$ = x$$_{2}$$, y$$_{3}$$ = x$$_{3}$$, etc.,

Thus, from the above explanation we conclude that for any point (x, y) on the line,

y-coordinate = x-coordinate

i.e., x = y, where (x, y) is any point on the line.

y = x is the equation of the straight line PQ.

Definition of the equation of a straight line:

The equation of a straight line is the common relation between the x-coordinate and y-coordinate of any point on the line.

Note: The coordinates of any point on the straight line satisfy the equation of the line.

Let the equation of a straight line y = 5x - 2. The point (1, 3) lies on the line y = 5x- 2 because (1, 3) satisfy the equation y = 5x – 2. Since by plugging 1 for x and 3 for y in the equation, we get 3 = 5(1) – 2 i.e., ⟹ 3 = 5 – 2 ⟹ 3 = 3, which is true.

But the point (2, 4) does not a lies on the line y = 5x- 2 because (2, 4) does not satisfy the equation y = 5x – 2.

Since by plugging 2 for x and 4 for y in the equation, we get 4 = 5(2) – 2 i.e., ⟹ 4 = 10 – 2 ⟹ 4 = 8, which is not true.

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Equation of a Straight Line