EXPLANATION.

Equation of the line through the point of intersection of lines,

x + 3y = 8  and  x + y = 1 and parallel to the line 5x - 7y = 3.

⇒ x + 3y = 8. - - - - - (1).

⇒ x + y = 1. - - - - - (2).

Subtracting equation (1) and equation (2), we get.

⇒ (x + 3y) - (x + y) = 8 - 1.

⇒ x + 3y - x - y = 7.

⇒ 3y - y = 7.

⇒ 2y = 7.

⇒ y = 7/2.

Put the value of y = 7/2 in equation (2), we get.

⇒ x + y = 1. - - - - - (2).

⇒ x + 7/2 = 1.

⇒ x = 1 - 7/2.

⇒ x = (2 - 7)/(2).

⇒ x = - 5/2.

∴ value of x is - 5/2  and  y is 7/2.

Slope of parallel line : ax + by + c = 0 is - a/b.

Slope of parallel line : 5x - 7y - 3 = 0 is (-5/-7) = 5/7.

Slope : m = 5/7.

Equation of the line.

⇒ (y - y₁) = m(x - x₁).

Using this formula in this question, we get.

⇒ (y - 7/2) = (5/7)[x - (-5/2)].

⇒ (y - 7/2) = (5/7)(x + 5/2).

⇒ 7(y - 7/2) = 5(x + 5/2).

⇒ 7[(2y - 7)/(2)] = 5[(2x + 5)/(2)].

⇒ 7(2y - 7) = 5(2x + 5).

⇒ 14y - 49 = 10x + 25.

⇒ 10x + 25 - 14y + 49 = 0.

⇒ 10x - 14y + 74 = 0.

∴ The equation of line is 10x - 14y + 74 = 0.