Question

Solve each given equation and show your work. Tell whether it has one solution, an infinite number of solutions, or no solutions, and identify each equation as an identity, a contradiction, or neither.

(a) 6x+4x-6=24+9x
(b) 25-4x=15-3x+10-x
(c) 4x+8=2x+7+2x-20

Answer 1

(i) The value of x is 30. The equation has one solution and it is neither identity nor contradiction.

(ii) The equation has infinite number of solutions and it is identity.

(iii) The equation has no solutions and it is a contradiction.

Explanation:

If an equation have infinitely many solutions then it is called an identity equation.

If an equation that has no solution, then it is called a contradiction.

(i)

`6x+4x-6=24+9x`

Combine like terms.

`10x-6=24+9x`

Isolate variable terms.

`10x-9x=24+6`

`x=30`

The value of x is 30. The equation has one solution and it is neither identity nor contradiction.

(ii)

`25-4x=15-3x+10-x`

Combine like terms.

`25-4x=(15+10)+(-3x-x)`

`25-4x=25-4x`

Isolate variable terms.

`4x-4x=25-25`

`0=0`

This equation is true for all values of x.

Therefore, the equation has infinite number of solutions and it is an identity.

(iii)

`4x+8=2x+7+2x-20`

Combine like terms.

`4x+8=(2x+2x)+(7-20)`

`4x+8=4x-13`

Subtract 4x from both sides.

`4x+8-4x=4x-13-4x`

`8=-13`

This equation is not true for any value of x.

Therefore, the equation has no solutions and it is a contradiction.