**Problem 1 : **

Solve the following equation :

2(1 - x) + 5x = 3(x + 1)

**Problem 2 : **

Solve the following equation :

(1/2)(8y - 6) = 5y - (y + 3)

**Problem 3 : **

Solve the following equation :

(1/3)(9 - 6x) = 5 - 2x

**Problem 4 : **

(1/3)(15 - 6x) = 5 - ax

If the linear equation above is an identity, what is the value of a ?

**Problem 5 : **

4x + 13 = 7(x - 2) + bx

If the linear equation above has no solution, what is the value of b ?

**Problem 1 : **

Solve the following equation :

2(1 - x) + 5x = 3(x + 1)

**Solution : **

2(1 - x) + 5x = 3(x + 1)

Simplify both sides.

2 - 2x + 5x = 3x + 3

2 + 3x = 3x + 3

Subtract 3x from each side.

2 = 3

The above result is false. Because 2 is not equal to 3. Because the result we get at the last step is false, the given equation has no solution.

**Problem 2 : **

Solve the following equation :

(1/2)(8y - 6) = 5y - (y + 3)

**Solution : **

(1/2)(8y - 6) = 5y - (y + 3)

Simplify both sides.

4y - 3 = 5y - y - 3

4y - 3 = 4y - 3

Subtract 4y from each side.

-3 = -3

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.

**Problem 3 : **

Solve the following equation :

(1/3)(9 - 6x) = 5 - 2x

**Solution : **

(1/3)(9 - 6x) = 5 - 2x

Simplify.

3 - 2x = 5 - 2x

Add 2x to each side.

3 = 5

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.

**Problem 4 : **

(1/3)(15 - 6x) = 5 - ax

If the linear equation above is an identity, what is the value of a ?

**Solution : **

(1/3)(15 - 6x) = 5 - ax

Simplify.

5 - 2x = 5 - ax

Because the given equation is an identity, the coefficients of like terms on both sides must be equal.

That is, coefficients of 'x' terms on the left side and right side must be equal.

So, equate the coefficients of 'x'.

- 2 = - a

Multiply each side by (-1).

2 = a

If the given linear equation is an identity, the value of a is 2.

**Problem 5 : **

4x + 13 = 7(x - 2) + bx

If the linear equation above has no solution, what is the value of b ?

**Solution : **

4x + 13 = 7(x - 2) + bx

Simplify.

4x + 13 = 7x - 14 + bx

4x + 13 = bx + 7x - 14

4x + 13 = (b + 7)x - 14

If (b + 7) = 4, we have

4x + 13 = 4x - 14

Subtract 4x from each side.

13 = -14

The above result is false. Because 13 is not equal to -14.

If (b + 7) = 4, the result we get is false and the given equation has no solution.

Solve for b in 'b + 7 = 4'.

b + 7 = 4

Subtract 7 from each side.

b = -3

If the given linear equation has no solution, the value of b is -3.

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