Chapter 1.3, Problem 22E

To determine

To solve the equation:

$12y+6=6\left(2y+1\right)$

Also determine whether the equation has one solution, no solution or infinitely many solution.

Answer to Problem 22E

The equation has infinitely many solutions.

Explanation of Solution

Given:

$12y+6=6\left(2y+1\right)$

Concept Used:

• An equation that is true for all the values of variable has infinitely many solution. And an equation that is not true for any value of the variable has no solution.
• Steps to solve a linear equation:
1. Use distributive property to remove any grouping symbol.
2. Simplify the expression in each side of the equation.
3. Collect the variable terms on one side of the equation and the constant on the other side.
4. Isolate the variable.

Calculation:

In order to solve the equation:

$12y+6=6\left(2y+1\right)$

First, simplify the expression in each side of the equation, as

  $\begin{array}{c}12y+6=6\left(2y+1\right)\\ 12y+6=12y+6\end{array}$

Now, collect the variable terms on one side of the equation and the constant on the other side, as

$\begin{array}{c}12y+6=12y+6\\ \Rightarrow 12y-12y=6-6\\ \Rightarrow 0=0\end{array}$

Now, the statement $0=0$ is true for every value of variable y . Thus, the given equation is an identity.

Thus, the given equation has infinitely many solutions.