Format of the equation — ax + b = 0 …(1) Where a , b ∈ R & a≠0 it is known as a linear equation. Root of equation (1) : x= -b/a
Example 1. Solve the equation 2x + 3(x-1) = x+7 solution. We have 2x+3(x-1) = x+7
⇒ 2x+3x-3=x+7 ⇒ 5x-x=7+3 ⇒ 4x=10 ⇒ x=10/4 ⇒ x=5/2 Example 2. Solve the equation: (a-5)x + 13 = a+8
solution. We have (a-5)x + 13 = a+8
⇒ (a-5)x=(a-5)
Case1: for a≠5 ⇒ x=(a-5)/(a-5) ⇒ x=1 Case2: for a=5 ⇒ 0.x+13=13 ⇒ 13=13 Here any real number is its solution.
Example 3.In a triangle, one angle measure half the measure of the largest angle, and another angle measure 30 degrees more than the smallest angle. Find all angles of the triangle
Solution :
We know that
Sum of all angles in triangle = 180
Let largest angle = 2x
So , one angle is = 1/2(largest angle)
= 1/2(2x)
= x
And the third angle = smallest angle + 30
= x + 30
Then summation of all angles = 180
(x ) + (2x) +(x +30) = 180
4x + 30 = 180
4x = 180 – 30
4x = 150
x = 150/4
x = 37.5
And angle of triangle is ———
x = 37.5
2x = 75
x + 30 = 67.5
So , angles37.5 , 67.5 & 75 respectively
Example 4: solve the linear equations for x & y 4x + 7y = 15 3x + 2y = 10 Solution : Given equations – – – 4x + 7y = 15 – – – (1) 3x + 2y = 10 – – – (2) To Make the same coefficient of x : 3 [ 4x + 7y = 15 ] – – -(3) 4 [ 3x + 2y = 10 ] – – – (4) Then 12x +21y = 45 – – – (3) 12x + 8y = 40 – – – (4) Subtracting: Eq (3) -Eq(4)
13y= 5 y = 5/13 Putting value y = 5/13 in equation(2) Then 3x = 10 -2(5/13) = (130 – 10)/13 2x= 120/13 x = 60 / 13