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.






10th Grade Math

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