Linear equation

Linear Equation:

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 , angles  37.5 , 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

so , x & y is 5/13 & 60/13 respectively